Riyadh Final

Crosstalk and Signal Integrity in Ring Resonator
Based Optical Add/Drop Multiplexers for

Wavelength-Division-Multiplexing Networks
by
Riyadh Dakhil Mansoor , B.Eng, M.Sc, MIET.
School of Engineering and Sustainable Development
De Montfort University
A thesis submitted in partial fulfilment of the requirements for the degree of

Doctor of Philosophy
July 2015
To my family.
i
ABSTRACT
With 400 Gbps Ethernet being developed at the time of writing this thesis, all-optical
networks are a solution to the increased bandwidth requirements of data communication
allowing architectures to become increasingly integrated. High density integration of
optical components leads to potential ‘Optical/Photonic’ electromagnetic compatibility
(EMC) and signal integrity (SI) issues due to the close proximity of optical components and
waveguides. Optical EMC issues are due to backscatter, crosstalk, stray light, and substrate
modes. This thesis has focused on the crosstalk in Optical Add/Drop Multiplexers
(OADMs) as an EMC problem.
The main research question is: “How can signal integrity be improved and crosstalk effects
mitigated in small-sized OADMs in order to enhance the optical EMC in all-optical
networks and contribute to the increase in integration scalability?” To answer this question,
increasing the crosstalk suppression bandwidth rather than maximizing the crosstalk
suppression ratio is proposed in ring resonator based OADMs. Ring resonators have a small
‘real estate’ requirement and are, therefore, potentially useful for large scale integrated
optical systems.
A number of approaches such as over-coupled rings, vertically-coupled rings and rings with
random and periodic roughness are adopted to effectively reduce the crosstalk between 10
Gbps modulated channels in OADMs. An electromagnetic simulation-driven optimization
technique is proposed and used to optimize filter performance of vertically coupled single
ring OADMs. A novel approach to analyse and exploit semi-periodic sidewall roughness in
silicon waveguides is proposed. Grating-assisted ring resonator design is presented and
optimized to increase the crosstalk suppression bandwidth.
ii
Approval Page
This is to certify that the work in this thesis consists of original work undertaken solely by
myself. Information taken from the published work of others has been properly referenced.
The material described in this thesis has not been submitted for the award of a higher
degree or qualification in any other university.
Riyadh Mansoor
Leicester, 2015
iii
ACKNOWLEDGEMENTS
First and foremost, I am grateful to Almighty God for enabling me to continue with this
hard journey.
A debt of gratitude must be paid to my supervisor Prof. Alistair Duffy for his support
during the course of my PhD and for knowing when to direct me and when to let me go my
own way.
I would also like to thank Dr. Hugh Sasse, Dr. Mohammed Al-Asadi and Dr. Stephen Ison
for their good humoured assistance throughout the course of the project. I want to extend
my thanks to Prof. Slawomir Koziel from Reykjavik University for his wonderful
collaboration and support for the optimization of vertical model. Thank and gratitude to
Prof. Melloni and Dr. Daniele from the Politecnico di Milano/ Italy for sharing their
simulator for validation.
I gratefully acknowledge the funding source that facilitated my PhD work. I would like to
express my deep gratitude to the Ministry of Higher Education and Scientific Research/
Iraq, the Iraqi cultural attaché /London and the Ministry of Industry and Minerals /Iraq for
their support.
Last, but not the least, I am very thankful to my parents, my wife (Shafaq) and my lovely
kids for their support and patience throughout the duration of my PhD.
iv
TABLE OF CONTENTS
ABSTRACT .............................................................................................................................. II
ACKNOWLEDGEMENTS ......................................................................................................... IV
LIST OF TABLES .................................................................................................................... IX
LIST OF FIGURES ................................................................................................................... X
LIST OF ABBREVIATIONS ....................................................................................................XIV
LIST OF PUBLICATIONS ......................................................................................................XVI
CHAPTER ONE INTRODUCTION ....................................................................... 1
1.1. Motivations 1
1.2. All-Optical Networks 6
1.3. Aims and Objectives 9
1.4. New Contributions to Knowledge 10
1.5. Outline of the Thesis 11
CHAPTER TWO CROSSTALK IN ALL-OPTICAL NETWORKS .................. 12
2.1. Introduction 12
2.2. Optical Crosstalk 14
2.2.1. Classification of Crosstalk ........................................................... 14
2.2.2. Modelling of Crosstalk ................................................................ 17
2.3. Crosstalk In Optical Add/Drop Multiplexers 19
2.3.1. Array Waveguide Grating Based OADM ................................... 20
2.3.2. Fibre Bragg Grating Based OADM ............................................. 21
v
CONTENTS Page
2.3.3. Ring Resonator Based OADM .................................................... 23
2.4. Crosstalk Modelling in Ring Resonator Based OADMs 24
2.4.1. Inter-Band Crosstalk .................................................................... 24
2.4.2. Intra-Band Crosstalk .................................................................... 26
2.5. Mitigation of Crosstalk in Ring Resonator Based OADMs 30
2.6. Conclusion 34
CHAPTER THREE OPTICAL RING RESONATORS ........................................... 35
3.2. Ring Resonators 37
3.3. SOI Strip Waveguides 42
3.3.1. Dispersion, Effective Refractive Index and Group Index ........... 43
3.3.2. Directional Coupler ..................................................................... 44
3.4. CST Microwave Studio 47
3.5. Coupling in Ring Resonators 49
3.5.1. Lateral Coupling .......................................................................... 49
3.5.2. Vertical Coupling ........................................................................ 50
3.6. Cascaded Coupled Ring Resonators. 51
3.6.1. Series Coupling ........................................................................... 51
3.6.2. Parallel Coupling ......................................................................... 53
3.7. Optical Add/Drop Multiplexer 54
vi List of Contents
CONTENTS Page
3.8. Conclusion 59
CHAPTER FOUR OVER-COUPLED RING RESONATOR BASED OADM .... 60
4.2. Crosstalk Bandwidth in a Single Ring Resonator 63
4.2.1. CST Simulation ........................................................................... 64
4.2.2. Analytical Calculations ............................................................... 65
4.3. Crosstalk Bandwidth in Double Ring Resonator 70
4.4. Conclusion 79
CHAPTER FIVE CROSSTALK BANDWIDTH ESTIMATION OF

PARALLEL COUPLED OADM ........................................................... 81
5.2. Mason’s Rule for Parallel Coupled Ring Resonators 83
5.3. CST Simulation 91
5.4. Analytical and Simulation Results 94
5.5. Conclusion 99
CHAPTER SIX VERTICALLY COUPLED RING RESONATOR OADM ...... 101
6.2. Crosstalk Suppression: Analytical and Simulation Model 103
6.2.1. Analytical Model ....................................................................... 103
6.2.2. CST Simulation ......................................................................... 106
6.3. Crosstalk Suppression Bandwidth 109
6.4. Optimization Method 110
vii List of Contents

CONTENTS Page
6.5. Conclusion 116
CHAPTER SEVEN GRATING-ASSISTED RING RESONATOR OADM ........ 117
7.2. Coupled Mode Analysis 120
7.2.1. Time Domain Analysis .............................................................. 120
7.2.2. Space Domain Analysis ............................................................. 125
7.3. CST Validation 129
7.4. Controllable Reflectivity 134
7.5. Grating-Assisted Single Ring 140
7.6. Conclusion 144
CHAPTER EIGHT CONCLUSIONS AND FUTURE WORK ............................. 145
8.1. Conclusions 145
8.2. Suggestions for Future Work 150
REFERENCES ...................................................................................................................... 152

APPENDICES ....................................................................................................................... 173
Appendix A: SFG METHOD FOR PARALLEL COUPLED OADMS ........ 173
Appendix B: RUBY CODE FOR SIDEWALL ROUGHNESS GENERATION . 176
viii List of Contents

LIST OF TABLES
TABLE Page
4-1. The relation between the inner and outer coupling coefficients for optimum
coupling. ............................................................................................................ 71
4-2. Inter-ring coupling coefficients for over-coupling. ........................................... 77
6-1. Optimization results ........................................................................................ 113
ix
LIST OF FIGURES
FIGURE Page
1-1. All-optical communication [27]..................................................................... 7

1-2. Size reduction of PLC’s (series coupled ring resonator (left), PLC chip
(centre) and a silicon wafer with hundreds of chips (right) [6]. .................... 8
2-1. An explanation of crosstalk components in the cross connector [56]. ........ 16
2-2. Crosstalk in OADM. .................................................................................... 16
2-3. a. Array Waveguide Grating. b. The N channels AWG based OADM
[66]. .............................................................................................................. 21
2-4. Fibre Bragg Grating based OADM [54]. ..................................................... 22
2-5 a. Ring resonator add/drop filter. b. Racetrack resonator based
OADM. ........................................................................................................ 23
2-6. Drop port response for single (solid) and double (dashed) ring
resonators. .................................................................................................... 26
2-7. Crosstalk suppression of a single ring resonator OADM. ........................... 28
2-8. Power penalty as a function of crosstalk suppression ratio. ........................ 29
3-1. A schematic diagram of a ring resonator based all-pass filter. .................... 38
3-2. Different geometries of SOI directional couplers. ....................................... 45
3-3. CST simulations of the fundamental TE like mode distribution at the
input port of a single ring OADM ............................................................... 49
3-4. Laterally coupled ring resonator. ................................................................. 50
3-5. Vertical coupled ring resonator.................................................................... 51
3-6. The schematic of N-series coupled ring resonator....................................... 53
3-7. The schematic of N-parallel coupled ring resonator. ................................... 54
3-8 Schematic diagram of the ring resonator based OADM. ............................. 55
3-9 CST simulated frequency response of a single ring resonator based
OADM. ........................................................................................................ 59
x
FIGURE Page
4-1. Analytically calculated drop port response of a series coupled RR,

with a study of the inter-ring coupling (𝒌𝒊 𝟐 ) effect on the resonance
splitting. ....................................................................................................... 62
4-2. Schematic of a single ring add/drop filter with three NRZ of 10 Gbps
modulated WDM signal. .............................................................................. 64
4-3. Frequency response of a single ring resonator. ............................................ 65
4-4. Bandwidth of crosstalk suppression and DPRR for a single ring
resonator....................................................................................................... 67
4-5. Bandwidth for asymmetric single ring resonator as a function of
coupling coefficients. ................................................................................... 69
4-6. Bandwidth of a symmetric single ring resonator as a function of
coupling coefficient and losses. ................................................................... 69
4-7. The schematic of a series double ring resonator add/drop filter with 10
Gbps RZ WDM signal. ................................................................................ 72
4-8. a) Frequency response for series double ring resonator. b) Spectrum
splitting at resonance for series double ring resonator. ............................... 75
4-9. Bandwidth of crosstalk suppression and drop port rejection ratio for
critical and over coupled double ring resonator (losses =4 dB/cm). ........... 78
4-10. Bandwidth of the over coupled ring resonator as a function of losses
for different coupling coefficient. ................................................................ 79
5-1. CST model of parallel coupled ring resonator. ............................................ 84
5-2. Forward paths. ............................................................................................. 86
5-3. Closed loops. ................................................................................................ 87
5-4. Analytically calculated spectral response for a 0.05 coupling. .................... 90
5-5. The model sensitivity for different coupling coefficients. ........................... 91
5-6. CST simulation of a symmetric parallel coupled ring resonator for
optimum separation (a) and half resonator length (b).................................. 93
5-7. The OBRR sensitivity to the coupling coefficients. .................................... 95
5-8. Coupling coefficient effects on the level of crosstalk suppression. ............. 95
xi List of Figures
FIGURE Page
5-9. Crosstalk as a function of losses. ................................................................. 96

5-10. Crosstalk as a function of coupling coefficients. ......................................... 97
5-11. Crosstalk bandwidth for an optimal coupling coefficient of 0.05,
calculated using CST MWS. ........................................................................ 98
5-12. Crosstalk bandwidth for a 0.06 coupling coefficient calculated using
CST MWS.................................................................................................... 99
6-1. (a) Vertical coupled ring resonator. (b) Cross section of the bent and
bus waveguides. ......................................................................................... 104
6-2. CST model of a vertically coupled ring resonator OADM. ....................... 106
6-3. The spectral response of CST simulated ring resonator OADM. .............. 107
6-4. Through port, Drop port, and out of band rejection ratio (OBRR) of
the ring resonator as a function of vertical separation for
optimized offset (a). ................................................................................... 108
6-5. Analytically calculated crosstalk suppression bandwidth as a function
of coupling coefficient. .............................................................................. 110
6-6. Initial (thin lines) and optimized (thick lines) responses for design
case (i) (two design variables): (a) |S21| and |S31| for 190.5 to 193.5
THz range, (b) magnification around 191 THz. Optimized 20dB-
bandwidth is 19.8 GHz. ............................................................................. 114
6-7. Initial (thin lines) and optimized (thick lines) responses for design
case (ii) (four design variables): (a) |S21| and |S31| for 190.5 to 193.5
THz range, (b) magnification around 191THz. Optimized 20dB-
bandwidth is 21 GHz. ................................................................................ 115
7-1. Forward and backward modes in a rough-walled ring resonator
add/drop filter. ........................................................................................... 121
7-2. The schematic diagram of a rough-walled ring resonator (a), and its
equivalent structure (b). ............................................................................. 126
xii List of Figures

FIGURE Page
7-3. a. Ring resonator response analytically modelled using time and space
models. b. Experimental (line) and analytical (dot) results presented in
[111]. .......................................................................................................... 129
7-4. CST model of sidewall roughness in a single ring resonator add /drop
filter. ........................................................................................................... 130
7-5. CST frequency response of a rough-walled Ring Resonator. .................... 131
7-6. CST (solid) and analytically (dotted) modelled spectral response for a
rough-walled ring resonator. ...................................................................... 132
7-7. The effect of back-reflection coefficient on the through and drop port
response. .................................................................................................... 133
7-8. Single (a), and double (b) gratings. ............................................................ 134
7-9. Grating length effect on the reflectivity for L1=6500 nm, L2= 13000
nm and L3= 19500 nm. .............................................................................. 136
7-10. The grating period effect on the reflectivity. ............................................. 137
7-11. Changing the reflectivity with increasing the number of gratings. ............ 138
7-12. Aspic model for three gratings (a), and the reflectivity as a function of
wavelength (b) for single grating (blue), double gratings (green) and
three gratings (red). .................................................................................... 139
7-13. The effect of separation between gratings for three gratings. .................... 140
7-14. a. The spectral response of a single ring resonator (using optimized
parameters that maximize crosstalk bandwidth), b. Schematic of a
grating-assisted OADM. ............................................................................ 143
7-15. The three port response for a grating assisted ring resonator (ASPIC
simulated results). ...................................................................................... 143
xiii List of Figures

LIST OF ABBREVIATIONS
AWG Array Waveguide Grating.
BER Bit-Error-Rate.
CW Continuous Wave.
EM Electromagnetic.
EMC Electromagnetic Compatibility.
FBG Fibre Bragg Grating.
FD Frequency Domain.
FSR Free Spectral Range.
FWHM Full Width at Half Maximum.
InP Indium Phosphide.
NRZ Non return-to-zero pulse.
OADM Optical Add/Drop Multiplexer.
OEO Optical –Electronic-Optical conversion.
OXC Optical cross-connector.
PLCs Planar Lightwave Circuits.
Q-factor Quality factor.
xiv List of Abbreviations

ROAD Reconfigurable Optical Add/Drop Multiplexer.
RZ Return-to-zero pulses.
SI Signal Integrity
Si Silicon.
SiO2 Silicon Dioxide.
SOA Semiconductor Optical Amplifier.
SOI Silicon On Insulator.
TD Time Domain.
TE Transverse electric.
TM Transverse magnetic.
WDM Wavelength Division Multiplexing.
xv List of Abbreviations
LIST OF PUBLICATIONS
1. Journal papers
[P1] R. D. Mansoor, H. Sasse, M. A. Asadi, S. J. Ison and A. Duffy, “Over Coupled Ring
Resonator-Based Add/Drop Filters,” Quantum Electronics, IEEE Journal Of, vol. 50,
pp. 598-604, 2014.
[P2] R. D. Mansoor, H. Sasse, S. Ison and A. Duffy, “Crosstalk bandwidth of grating-

assisted ring resonator add/drop filter,” Optical and Quantum Electronics, vol.47, no.5,
pp.1127-1137, 2015.
[P3] R. D. Mansoor, S. Koziel*, H. Sasse, and A. Duffy,“Crosstalk Suppression Bandwidth

Optimization of a Vertically Coupled Ring Resonator Add/Drop Filter,” IET
Optoelectronics, vol.9, no.2, pp.30-36, April, 2015.
[P4] R. D. Mansoor, H. Sasse, M. A. Asadi, S. J. Ison and A. Duffy, “Estimation of the

Bandwidth of Acceptable Crosstalk of Parallel Coupled Ring Resonator Add/Drop
Filters,” Transactions on EMC, IEEE Journal of, June, 2015. DOI:
10.1109/TEMC.2015.2432914
[P5] R. D. Mansoor, H. Sasse and A. Duffy, “Optimization of Reflection Coefficient in

Ring Resonator Add/Drop Filters”, International Journal of Numerical Modelling:
Electronic Networks, Devices and Fields, 2015. DOI: 10.1002/jnm.2080.
[P6] R. D. Mansoor, H. Sasse and A. Duffy, “Modified Crosstalk suppression bandwidth

Single Ring Resonator Optical Add/Drop filter”, submitted for publication.
 S. Koziel is a professor of engineering optimization at Reykjavik University; he helped with the

preparing of the objective function and performing the optimization using fast computers in his lab.
H.Sasse provided technical and research support. Other co-authors are with the supervision team.
xvi
2. Conference papers
[P7] R. D. Mansoor, S. Ison, H. Sasse and A. P. Duffy, “Impact of crosstalk in all optical
networks,” Proceedings of the 61st IWCS Conference, pp. 849-855, Rhode Island,
USA, 2012.
[P8] R. D. Mansoor, H. Sasse and A. P. Duffy, “Analysis of Optical Ring Resonator

Add/Drop Filters,” Proceedings of the 62nd IWCS Conference, pp. 471-475, Charlotte,
USA, 2013.
[P9] R. Mansoor, H. Sasse, S. Ison, and A. Duffy, “Modelling of Back Reflection in

Optical Ring Resonators,” IEEE International Conference on Numerical
Electromagnetic Modelling and Optimization for RF, Microwave, and Terahertiz
Applications (NEMO), Pavia, Italy, 2014.
[P10] R. Mansoor, H. Sasse, and A. Duffy, “Optimization of Vertically Coupled Add/Drop

Ring Resonator Based Filter,” Proceedings of the Semiconductor and Integrated
OptoElectronics (SIOE) Conference, Cardiff, UK, 2014.
[P11] R. Mansoor, H. Sasse, and A. Duffy, “Modelling of A Roughened Sidewall Ring

Resonator Add/drop Filter,” Proceedings of the XXII International Workshop on
Optical Wave & Waveguide Theory and Numerical Modelling (OWTNM), Nice,
France, 2014.
[P12] R. Mansoor, H. Sasse, and A. Duffy, “Enhancing the depth notch using a rough-
walled SOI ring resonator” IEEE Optical Interconnects, San Diego, California, USA,
2015.
[P13] R. Mansoor, A. Duffy, “Review of Progress in Optical Ring Resonators with

Crosstalk Modelling in OADMS” the 64th IWCS conference, Atlanta, USA, 2015.
[P14] R. Mansoor, M. Al-Asadi, and A. Duffy, “Optical Ring Resonator Add/Drop
Filters”, Derby Electrical and Electronic Research Showcase (DEERS), Derby, UK,
2015.
xvii List of Publications

CHAPTER ONE
INTRODUCTION
This chapter introduces the basic definition of ‘Optical/Photonic’ EMC and explains why it
is of relevance in all-optical networks. Issues related to the performance of optical nodes
and integration solutions are discussed; research outcomes and an outline of the rest of the
thesis are provided.
1.1. Motivations
Optical Ethernet is a combination and extension of two existing technologies, Ethernet and
Optical communication technology [1]. Ethernet started in 1973 with the aim of connecting
personal computers, printers, and servers within copper local area networks LANs [2]. The
optical Ethernet in the First Mile (EFM) was introduced in IEEE standard 802.3ah in 2001
for 1 Gbps transmission [3]. The Ethernet Passive Optical Network (EPON) was launched
in 2010 as IEEE Standard 802.3av for 10 Gbps [4]. The IEEE Standard Project P802.3bs
extending the bandwidth to 400 Gbps was proposed in 2013 and was intended to meet the
increased requirements of multimedia communications and other data-dense applications
such as large CAD file sharing [5].
1
CHAPTER ONE INTRODUCTION
Nowadays, telecommunications companies are increasingly focusing on more integrated
solutions to cope with increased data transmission [6]. However, the integrated solutions to
meet optical network requirements are subject to many factors that affect signal integrity,
such as crosstalk and backscattering [7-10]. In view of this trend in all-optical networks,
where competition in integration density and signal integrity is the driving force, there will
be an increasing demand to address these issues of Optical/Photonic “EM Compatibility”
especially in large scale integration technologies.
Electromagnetic Compatibility (EMC) is frequently described as being a “DC to light”
phenomenon. EMC is defined by the International Electrotechnical Commission IEC as
"The ability of a device, unit of equipment or system to function satisfactorily in its
electromagnetic environment without introducing intolerable electromagnetic disturbances
to anything in that environment" [11]. However, it is only with the move to increasingly all-
optical networks that EMC of optical communications systems becomes a topic worth
special attention in its own right. In this research, EMC is of relevance because add/drop
multiplexers and filters are used to introduce and remove channels in Wavelength-Division
Multiplexing nodes. Non-ideal behaviour can lead to the introduction of troublesome
crosstalk between the dropped and added channels, with resulting increases in Bit Error
Rate (BER) and loss of signal integrity [12]. By definition, “integrity” means “unimpaired
and complete”. Therefore, an optical signal with good integrity has a clean spectrum with a
high suppression for unwanted channels.
2
Optical EMC issues are due to crosstalk, backscatter, stray light, and substrate modes that
can no longer be regarded as negligible, especially in large scale integration technologies.
Photonic integrated circuits are fabricated by combining optical waveguides closely in a
single chip. Interconnections (optical waveguides) affect signal integrity due to light
leakage between waveguides and are becoming a serious problem [13, 14]. Therefore, a
high level of isolation is needed since all channels are processed simultaneously on chip
[15], and just as in electronics, EMC issues that can disturb the operation of the PLC should
be considered [7].
In this thesis, the topic of “Optical” EMC is approached by studying the crosstalk in all-
optical networks; in the particular case of optical ring resonator based add/drop
multiplexers (OADMs) and filters. Ring resonator based OADMs and filters are playing an
important role in increasing the flexibility of Wavelength-Division Multiplexing (WDM)
networks by allowing the insertion and dropping of wavelength channels [16, 17].
Ring resonators are key components in modern all-optical networks [18]. Their small size
allows high density integration in optical/ Photonic circuits due to the use of high index
contrast materials and the availability of CMOS fabrication facilities [19, 20]. Coupling a
closed loop resonator with bus waveguides through the evanescent mode coupling leads to
filter behaviour of the resulting structure. However, like any other optical filters, ring
resonator based OADMs are prone to crosstalk. Inter and Intra-band crosstalk in ring
resonator based OADMs result from the non-ideal dropping of channels [21]. The dropped
3
channel will be corrupted by the residual of a new added channel (intra-band crosstalk).
Inter-band crosstalk also occurs due to the adjacent channels in a WDM signal [22].
Crosstalk in ring resonator based OADMs was mitigated by improving filter response using
high order filters and increased notch depth in the through port response [23-26]. Increasing
filter order (number of rings) or connecting different OADMs, giving a multi-stage
structure, results in a reduction in the inter-band crosstalk. However, filter size will
increase, conflicting with the goal of greater device density. While, increasing the notch
depth by optimizing coupling coefficients (to mitigate the intra-band crosstalk) will only
increase the crosstalk suppression ratio in a narrow band of frequencies at resonance. For
modulated channels this implies that the side-bands will get different levels of suppression
from that of the centre frequency. Each channel in WDM networks is modulated with
different information [27]. For example, for 10 Gbps non return-to-zero (NRZ)
transmission, the required bandwidth is 20 GHz [24]. Therefore, it is required to maintain a
high level of crosstalk suppression ratio for a 20 GHz bandwidth in order to maintain signal
integrity of the added/dropped channels.
Based on the calculations in [23, 28] it was shown that a level of |20| dB for the crosstalk
suppression ratio represents a sufficient level for acceptable BER and signal integrity.
Therefore, this thesis suggests that increasing the crosstalk suppression bandwidth will
allow for adding/dropping of modulated channels with equal level of crosstalk suppression
over the whole side-bands and leading to an effective reduction in the crosstalk between the
4
channels in OADMs and filters. The crosstalk suppression bandwidth is defined as the
bandwidth over which the level of crosstalk suppression is maintained at ≥ |20| dB.
In this thesis, a number of approaches are adopted in order to increase the crosstalk
suppression bandwidth in small-sized OADMs. Analytical and numerical models are used
to optimize coupling coefficients in laterally coupled rings (series and parallel double rings)
and a vertically coupled (single) ring resonator based OADMs in order to achieve high
crosstalk suppression bandwidth. Sidewall roughness in a single ring resonator is also
exploited to improve signal integrity by increasing the crosstalk suppression bandwidth. A
proper investigation of these methodologies is presented in the following chapters.
Hence, the motivations behinds this research can be summarised as follows:
1. Optical communication is an extremely fast growing technology, driven mainly by the
increasing need for global expansion of the internet and, in particular, multimedia
communications and other data-dense applications.
2. The need for high integration density and fabricating small components with high
scalability to cope with increased demand for data transmission.
3. The integrated solutions to meet optical network requirements are prone to many factors
that affect signal integrity, such as backscatter, crosstalk, stray light, substrate modes.
The optical/Photonics community deals with many of the key issues, but do not
specifically think of this as EMC and the EMC community has studied many of these
issues, but not at optical frequencies. So, while this thesis presents a number of novel
5
contributions to the technology, it also aims to help provide a link between these
discipline areas.
1.2. All-Optical Networks
Fibre optic communication systems can exploit the large bandwidth of optical fibres
defined by the low propagation loss [29, 30] (14 THz at 1.33 µm and 15 THz at 1.55 µm).
However, using a single wavelength channel to exploit this huge bandwidth is impractical
due to amplifying and switching bottlenecks. The advent of Wavelength-Division
Multiplexing (WDM) has had a major impact on the evolution of high transmission rates
and low cost networks [31, 32]. WDM is a technique that employs many closely spaced
optical-frequency wavelengths to transmit multiple data signals [27, 33].
In WDM networks, a number of nodes are required to provide switching, routing, and
adding/dropping of channels [34]. Optical nodes in the first generation of WDM networks
were based on Optical-Electrical-Optical (OEO) conversion for each wavelength,
regardless of whether the wavelength would be dropped at this node or would pass through
[29, 35]. The drawbacks of the OEO process are the cost, space requirements, and power
consumption, especially for long distances and high capacity networks [31].
The fabrication of new optical components such as Optical Add/Drop Multiplexers
(OADM) [16], Optical Cross-Connecters (OXC) [36] and amplifiers [37] had led to the
second generation of WDM networks (All-optical or transparent networks) as shown in
Figure 1-1. All-optical networks have emerged as a solution to keep up with the increasing
6
throughput demand. With all-optical networks, the transmitted signals are treated
completely in the optical domain [38]. Therefore, bottlenecks presented by OEO
conversion have been eliminated, allowing for higher data rate transmission using different
signal formats [33, 39].
Figure 1-1. All-optical communication [27].
Optical nodes in the early all-optical networks were realized by using a hybrid technology,
where each component was fabricated separately and then connected together [40, 41]. This
resulted in large size and high cost optical nodes. The second generation of optical nodes
combined a high number of optical functions in a single integrated device. A similar
approach to that used for electronic integrated circuits was suggested [42]. This approach
had a lot of advantages over the hybrid one, especially in terms of functionality and cost
reduction. A number of devices per chip were fabricated [43, 44]. The drawback of this
7
technology was the integration complexity and the difficulty of scaling to meet the network
growth.
Planar Lightwave Circuits PLCs were introduced [6] to reduce cost and complexity and, at
the same time, improve the scalability of optical networks. Silicon-On-Insulator (SOI)
technology [45] offers strong confinement of light in a small size optical waveguide, which
makes it of great interest in the fabrication of PLCs. Figure 1-2 shows the size reduction
obtained with PLCs made using SOI and explains how a large number of optical devices
are integrated in a single silicon wafer. However, the increasing levels of integration,
greater data rate and greater bandwidth requirements mean that the limiting factors are
getting similar to those faced in electronic circuits and, hence, “Photonic EMC” is worthy
of study and interest.
Figure 1-2. Size reduction of PLC’s (series coupled ring resonator (left), PLC chip (centre) and a silicon
wafer with hundreds of chips (right) [6].
8
1.3. Aims and Objectives
 This thesis aims to:
1. Introduce “Photonic” EM Compatibility.
2. Mitigate the crosstalk effect and improve signal integrity in ring resonator based
OADMs by increasing the crosstalk suppression bandwidth.
3. Reduce filter size and contribute to the increase in integration scalability.
 The objectives are:
1. To exploit the resonance splitting that occurs due to the inter-ring coupling coefficients
in a series coupled OADM.
2. To enhance the crosstalk suppression bandwidth by optimizing ring parameters in a
parallel coupled OADM.
3. To optimize ring parameters of a vertically coupled OADM in order to increase the
crosstalk suppression bandwidth in a small-sized OADM.
4. To exploit the resonance splitting induced by sidewall roughness in rough-walled ring
resonators to maximize the crosstalk suppression bandwidth.
By following the above objectives, a number of novel designs and models are obtained, as
listed in the next section.
9
1.4. New Contributions to Knowledge
The main contributions of this thesis are as follows:
1. The over-coupling condition between the inter-rings in series coupled ring resonators
OADMs is proposed. Over-coupling, simultaneously, improves the bandwidth and the
level of inter-band crosstalk suppression, and allows for high data rate signal dropping
compared to critically coupled OADMs.
2. A general form of the transfer function of parallel coupled ring resonator OADMs is
derived using the Signal Flow Graph method (based on Mason’s rule). The use of
Mason’s rule in this regard is novel and useful. The derived model provides an accurate
starting point for design and analysis and, in doing this, provides a better insight into
the filter performance.
3. An electromagnetic simulation-driven optimization technique is proposed and used to
improve the crosstalk suppression bandwidth of a vertically coupled ring resonator
based OADM.
4. A novel design of grating-assisted ring resonator is proposed to increase the crosstalk
bandwidth compared to the smooth-walled ring OADM. A general model for a rough-
walled ring resonator is derived using the time and space domain Coupled Mode
Theory.
10
1.5. Outline of the Thesis
The remainder of this thesis is organised as follows:
Chapter 2 investigates crosstalk in all-optical networks; emphasis is given to the crosstalk
in ring resonator based OADMs.
Chapter 3 provides a general background of optical ring resonators and presents their
features in WDM networks.
Chapter 4 looks into crosstalk issues in series coupled ring resonator OADMs and
examines the crosstalk performance as a function of the inter-ring coupling coefficients.
Chapter 5 examines the performance of parallel coupled ring resonators OADMs and
focuses on the optical signal integrity perspective.
Chapter 6 proposes a design of small size vertically coupled ring resonator OADMs that
provides an increased crosstalk suppression bandwidth.
Chapter 7 proposes a general solution for rough-walled ring resonators modelling, as well
as a particular solution to maximize the crosstalk suppression bandwidth.
Chapter 8 concludes the thesis with a summary of its main findings and recommendations
for future work.
11
CHAPTER TWO
CROSSTALK IN ALL-OPTICAL
NETWORKS
In this chapter, crosstalk and signal integrity issues in all-optical networks are introduced
and discussed. A mathematical definition of crosstalk in optical cross-connectors and
add/drop filters is presented. Two main topics are discussed: First, to increase the
integration density in optical integrated circuits, ring resonators are good candidates due
to their small size. Second, to improve the crosstalk performance in WDM networks, the
crosstalk suppression ratio should be kept ≥ |20| dB over a wide range of frequencies
(covering the entire bandwidth of modulated dropped channels). This frequency range will
be defined as the “crosstalk suppression bandwidth” throughout this thesis.
2.1. Introduction
Wavelength-Division Multiplexing (WDM) systems transmit a number of channels at data
rates of 10 Gbps or higher in each wavelength channel [27]. The number of transmitted
channels is limited by the bandwidth of the optical devices used, such as optical amplifiers.
In order to meet the required information capacity, these channels are allocated close to
12
CHAPTER TWO CROSSTALK IN ALL-OPTICAL NETWORKS
each other (within the optical amplifier bandwidth). Passing the closely-spaced wavelength
channels through optical devices such as filters and multiplexers will result in the
occurrence of system impairments (linear crosstalk) [46, 47]. Other impairments, however,
will occur due to the nonlinearity of the medium induced by the combined optical power of
individual channels (nonlinear crosstalk). As a result, optical crosstalk represents a major
limiting factor in WDM networks [48, 49].
Optical add/drop multiplexers OADMs and filters that drop one channel of WDM signal,
without disturbing other channels, are essential elements in all-optical networks [50]. Ring
resonator based OADMs are shown to be good candidates to realize integrated add/drop
filters for WDM networks [24]. However, ring resonator based OADMs are prone to
crosstalk. This chapter aims to:
1. Study the crosstalk in all-optical networks
2. Model the crosstalk in ring resonator based OADMs.
This chapter is organized as follows:
1. Crosstalk in all-optical networks is discussed and classified. A mathematical definition
of the corrupted channels is presented.
2. Different OADMs structures are discussed and compared in terms of their size and
crosstalk performance.
3. Crosstalk in ring resonator based OADMs is modelled and an overview of the current
state of knowledge about mitigating crosstalk is presented.
13
This chapter suggests that maintaining an adequate level of crosstalk suppression ratio for a
wide bandwidth (covering the side-bands of modulated channels) is of substantial
importance to improve the overall performance in WDM networks.
2.2. Optical Crosstalk
2.2.1. Classification of Crosstalk
Crosstalk in all-optical networks can be classified as linear or nonlinear depending on the
network topology and components used. In metropolitan and long-haul networks, optical
fibre characteristics are strongly affected by the power and frequency of propagated WDM
channels. Although the power in each channel of the WDM signal may be below that
needed to produce fibre non-linearity, the total power summed over all channels can
quickly become significant [51]. Nonlinearity causes inter-channel effects such as Self
Phase Modulation (SPM) [52], Cross Phase Modulation (XPM) and Four Wave Mixing
(FWM) [53].
In local area networks (LANs), which are used to transmit data over short distances, the
effect of nonlinearity is less challenging. Linear crosstalk is the dominant type. Linear
crosstalk results from non-ideal performance of WDM nodes. Two types of linear crosstalk
are defined depending on the spectral location of crosstalk channels with respect to the
pass-band of optical filters: In-band and out-of-band crosstalk [25]. Optical out-of-band
crosstalk usually results from channels with spectra located out of the optical filter pass-
band. It is also called inter-band crosstalk and appears between channels of different
14
wavelengths. The ability to suppress this type by using narrow-band filters makes it less
harmful [24].
In-band (also called intra-band) crosstalk results from the residual of closely spaced
channels (separated by ∆𝑓), where ∆𝑓 is within the optical filter bandwidth. This type of
crosstalk is more problematic and causes serious degradation of system performance due to
the difficulty of removing it using optical filters. Furthermore, it will propagate with WDM
channels along the network and the destructive effect of this type of crosstalk will be
accumulated within each node it passes through [33, 47].
At the receiver side, the electrical filter bandwidth (B) will determine whether the intra-
band crosstalk is electrically in-band (𝛥𝑓 ≤ 𝐵), or out-of-band (𝛥𝑓 > 𝐵). The major
limitation in the system performance in the presence of crosstalk will result mainly from
electrical in-band crosstalk [34]. If 𝛥𝑓 = 0 (the signal and crosstalk have the same nominal
frequency), then the crosstalk will be homodyne [47]. Otherwise, if the signal and crosstalk
have a closely spaced frequency, i.e. 𝛥𝑓 ≠ 0 but still less than the electrical bandwidth, the
crosstalk will be heterodyne [54].
Optical cross-connectors [36, 55] and OADMs are essential elements in all-optical
networks. Cross-connectors consist of channel selective components (multiplexers/de-
multiplexers) and switching elements. Crosstalk in optical cross-connectors results from
adjacent WDM channels (inter-band) and from the delayed version of the channel itself
after travelling through different paths in the switching matrix (intra-band) as shown in
15
Figure 2-1. In OADMs, intra-band crosstalk results from the new added channels, while the
inter-band crosstalk results from adjacent WDM channels as shown in Figure 2-2.
Figure 2-1. An explanation of crosstalk components in the cross connector [56].
Figure 2-2 Crosstalk in OADM.
16
2.2.2. Modelling of Crosstalk
In this section, an optical system that uses on/off keying transmission is studied for
crosstalk modelling [57]. The optical field can be considered as a continuous wave (CW) of
the form:
𝑬𝒔 (𝑡) = 𝒓̂𝑠 √𝑃𝑠 𝑒 (jωs t+j∅s (t)) 2-1)
where, 𝑃𝑠 the optical power, 𝒓̂𝑠 represents the state of polarization, ωs is the CW
frequency, and ∅s (t) is the instantaneous optical phase.
After the propagation of an optical signal in an all-optical network, many crosstalk terms
will disturb it. The corrupted an optical field at the receiver input will be a combination of
the desired signal and the intra-system noise and can be expressed as:
𝑬𝒑𝒉 (𝑡) = 𝑬𝒔 (𝑡) + ∑ 𝑬𝒌 (𝑡)

𝑘
= 𝒓̂𝑠 √𝑃𝑠 𝑏𝑠 (𝑡)𝑒 (jωs t+j∅s(t)) + ∑ 𝒓̂k √𝜖𝑘 𝑃𝑠 𝑏𝑘 (𝑡)𝑒 (jωkt+j∅k(t))

k=2 (2-2)
𝑃𝑘
where, 𝜖𝑘 = is the power ratio of the kth crosstalk component to the dropped channel
𝑃𝑠
power, and N is the total number of channels in a WDM signal. 𝑏𝑠 (𝑡) and 𝑏𝑘 (𝑡) represents
the binary symbols forming the amplitude modulating signal ∈ {0,1}.
The receiver in such systems consists of a photo-detector followed by an electrical filter
and a decision (threshold) circuit [58, 59]. The photo-detector output current is proportional
17
to the square of the incident optical field. The output of an electrical filter is then compared
with a decision threshold level (𝐼𝐷 ) to decide whether a “1” or “0” state was sent. The
power leakage from unwanted channels may lead to a “1” state at the receiver side while
the transmitted signal is “0” or vice versa, with resulting increases in Bit Error Rate (BER)
and loss of signal integrity.
The receiver photo-current can be written as:
𝑁 2
2
𝑖𝑝ℎ (𝑡) = 𝜌|𝑬𝑝ℎ (𝑡)| = 𝜌 |𝑬𝑠 (𝑡) + ∑ 𝑬𝑘 (𝑡)|
𝑘=2 (2-3)
where 𝜌 represents the photodiode responsitivity and it will be considered equal to unity
for simplicity. The general form of the photo-current will consists of four terms [58], as
below:
1. Signal power term [𝑃𝑠 𝑏𝑠 (𝑡)].
2. Crosstalk power term [ 𝑃𝑠 ∑𝑁

𝑘=2 𝑏𝑘 (𝑡) 𝜖𝑘 ].
3. Signal-crosstalk beating term

𝑁
[ 2𝑃𝑠 ∑ 𝒓̂𝑠 ⋅ 𝒓̂𝑘 √𝜖𝑘 𝑏𝑘 (𝑡)𝑏𝑠 (𝑡) cos[(𝜔𝑠 − 𝜔𝑘 )t + ∅𝑠 (t) − ∅𝑘 (t)]]

𝑘=2
4. Crosstalk-crosstalk beat noise term, which is of less importance in the study of the
system performance in terms of crosstalk impairment [60].
𝑁−1 𝑁
[ 2𝑃𝑠 ∑ ∑ 𝒓̂𝑘 ⋅ 𝒓̂𝑙 √𝜖𝑘 𝜖𝑙 𝑏𝑘 (𝑡)𝑏𝑙 (𝑡) cos((𝜔𝑘 − 𝜔𝑙 )𝑡 + ∅𝑘 (𝑡) − ∅𝑙 (𝑡))]

𝑘=2 𝑙=𝑘+1
18
The total current can be written as in (2-4) below:
𝑁 𝑁
𝑖𝑝ℎ (𝑡) = 𝑃𝑠 [𝑏𝑠 (𝑡) + ∑ 𝜖𝑘 𝑏𝑘 (𝑡) + 2 ∑ 𝒓̂𝑠 ⋅ 𝒓̂𝑘 √𝜖𝑘 𝑏𝑘 (𝑡)𝑏𝑠 (𝑡)
𝑘=2 𝑘=2
𝑁−1 𝑁
⋅ cos[(𝜔𝑠 − 𝜔𝑘 )t + ∅𝑠 (t) − ∅𝑘 (t)] + 2 ∑ ∑ 𝒓̂𝑘 . 𝒓̂𝑙 √𝜖𝑘 𝜖𝑙

𝑘=2 𝑙=𝑘+1 (2-4)
⋅ 𝑏𝑘 (𝑡)𝑏𝑙 (𝑡) cos[(ω𝑘 − ω𝑙 )𝑡 + ∅𝑘 (t) − ∅𝑙 (t)]]
The most important crosstalk contribution is the signal-crosstalk beating term which can be
considered as a random variable in term of the phase [61].
2.3. Crosstalk In Optical Add/Drop Multiplexers
Optical add/drop multiplexers OADMs can be found either in a fixed or reconfigurable
mode of operation [62]. In a Reconfigurable-OADM (ROADM), the dropped wavelength
can be adjusted based on the network requirements [63]. The crosstalk in ROADM results
from the presence of switches that perform the reconfiguration. Passive or fixed OADMs
are used to add/drop a preselected wavelength in the WDM node. No switches are required,
and each node will be used to add/drop a specific wavelength. Crosstalk in this
configuration results from the non-ideal separation between channels. Different types of
OADM are used in WDM nodes [64]. A comparison between the three major structures, in
terms of signal integrity and integration scalability, is presented in this section.
19
2.3.1. Array Waveguide Grating Based OADM
Array Waveguide Gratings (AWGs) [65], shown in Figure 2-3 a, are planar devices with an
array of waveguides that perform multiplexing and de-multiplexing of WDM channels.
Crosstalk in the AWG based OADM results from incomplete suppression of neighbouring
channels during de-multiplexing (in the first free propagation region− input slab) and
multiplexing (second free propagation region−output slab). An optical add/drop
multiplexer based on AWG is shown in Figure 2-3 b.
In Figure 2-3 b, 𝑚 channels are added/dropped and 𝑛 channels are passed (𝑚 + 𝑛 = 𝑁),
where N is the total number of channels in the input fibre. In this design, an AWG is used
to perform both de-multiplexing and multiplexing functions in the OADM [66]. In this
structure, each dropped channel λ𝑑 will be corrupted by three different crosstalk
contributions:
1. (𝑁 − 1) channels from the network (inter-band crosstalk)
2. 𝑚 −added channels (intra-band crosstalk).
3. 𝑛 −pass channels (this will be a combination of inter and intra-band crosstalk).
20
Figure 2-3. a. Array Waveguide Grating. b. The N channels AWG based OADM [66].
The delayed versions of the dropped channel, which are leaked to the 𝑛 pass signals, will be
added again with the dropped channel and this will lead to an increase in the intra-band
crosstalk [66]. The drawbacks of AWG based OADMs can be summarized as:
1. Integration scalability is limited by the leakage between channels due to the close
spacing between waveguides.
2. Crosstalk calculations in AWG [67, 68] have shown that the performance of
OADMs degrades as either or both 𝑚 or 𝑁 are increased.
2.3.2. Fibre Bragg Grating Based OADM
Another possible structure to perform add/drop functionality in WDM networks is the Fibre
Bragg Gratings based OADMs [69]. Bragg gratings are a periodic perturbation in the
effective refractive index of an optical waveguide [70]. Changing the refractive index will
21
make the waveguide acts as a filter centred on a specific wavelength. Therefore, a
wavelength that propagates in a FBG will be reflected back if it satisfies the Bragg
condition (𝜆𝐵𝑟𝑎𝑔𝑔 = 2 ⋅ 𝑛𝑒𝑓𝑓 ⋅ ʌ) [71], where, 𝑛𝑒𝑓𝑓 is the effective refractive index of the
fibre, and ʌ is the grating period. More discussion on Bragg gratings is presented in
Chapter seven.
Figure 2-4 shows a schematic diagram of a FBG based OADM [54]. Crosstalk in this
structure results from the leakage of added channels in the Bragg gratings which leads to
the presence of unwanted signals at the drop port [72]. Therefore, the dropped channel will
be a combination of the reflected channel (𝜆2 ) and the leakage of added channels.
Crosstalk in this structure depends on fibre Bragg reflectivity and, in general, it is less than
that on the AWG [73]. However, the optical circulators will increase the filter size and this
affects the integration scalability.
Figure 2-4. Fibre Bragg Grating based OADM [54].
22
2.3.3. Ring Resonator Based OADM
Figure 2-5 shows the schematic diagram of a first order (single) ring resonator based
OADM. In its simplest form, it consists of a pair of bus waveguides evanescently coupled
to a closed loop waveguide (ring or racetrack shape) [18]. More discussion on the ring
resonators will be presented in Chapter three.
a b
Figure 2-5 a. Ring resonator add/drop filter. b. Racetrack resonator based OADM.
Crosstalk in ring resonator based OADMs results from the non-ideal dropping of channels
[21]. The dropped channel will be corrupted by the residual of a new added channel (intra-
band crosstalk). Inter-band crosstalk also occurs due to the adjacent channels in a WDM
23
signal [22]. Ring resonators have a small ‘real estate’ requirement and are therefore
potentially useful for large scale integrated optical systems compared to their AWG and
FBG counterparts [74, 75].
2.4. Crosstalk Modelling in Ring Resonator Based OADMs
2.4.1. Inter-Band Crosstalk
Considering that the OADM (shown in Figure 2-5) is used to drop a single channel from a
WDM signal at the input port, the dropped channel will be corrupted by the residual of
(𝑁 − 1) adjacent channels. The drop port photo-current will consist of two terms (as in
(2-5)), 𝑖𝑠 is the receiver current due to the dropped channel, and 𝑖𝑛 is the summation of
crosstalk currents resulting from the leaked power of (N-1) adjacent channels.
Based on equation (2-4), equation (2-5) can be derived:
𝑁 (2-5)
𝑖𝑝ℎ = 𝑖𝑠 + 𝑖𝑛 = 𝑃𝑠 [𝑏𝑠 (𝑡) + ∑ 𝑏𝑘 (𝑡) 𝜖𝑘 ]
𝑘=2
The third and fourth terms of equation (2-4) are small and neglected due to the frequency
difference between the dropped and unwanted channels.
The level of crosstalk current, 𝑖𝑛 , depends on:
1. Bit pattern of the (𝑁 − 1) channels 𝑏𝑘 (𝑡). 𝑖𝑛 becomes maximum when all channels
are in a “1” state simultaneously, which represent the worst case.
2. Suppression ratio (𝜖𝑘 ) for each adjacent channel (drop port response).
24
Figure 2-6 shows the suppression ratio (𝜖𝑘 ) for three channels in a single and double ring
resonator based OADM separated by 50 GHz (as specified by the ITU-T G.694.1
telecommunication standards [22]).
For a single ring OADM, based on the drop port response (solid line in Figure 2-6), the
suppression ratio is:
1. For the first adjacent channel (50 GHz from the resonance), 𝜖𝑘 = −5 𝑑𝐵.
2. For the second channel (100 GHz from the resonance), 𝜖𝑘 = −10 𝑑𝐵.
3. For the third channel (150 GHz from the resonance), 𝜖𝑘 = −12 𝑑𝐵.
However, the drop port response depends on the coupling coefficients and also on the
number of resonators used (multiple rings). Increasing the order of the filter (the number of
rings) will lead to a sharp transition in the spectral response as shown in Figure 2-6 (dashed
line). Hence, a reduction in the inter-band crosstalk can be achieved. The suppression ratio
for adjacent channels in a double ring resonator OADM are: −14.5 , −26.5 and −32.7 dB
for 50, 100 and 150 GHz spaced channels, respectively. Reducing the effect of inter-band
crosstalk requires enhancing the drop port response shape by using multiple rings
(increasing the order of the filter) [76, 77], but at the expense of increasing filter size. More
discussion on cascaded ring OADMs is presented in Chapter three.
25
Figure 2-6. Drop port response for single (solid) and double (dashed) ring resonators.
2.4.2. Intra-Band Crosstalk
Intra-band crosstalk is the main source of system performance degradation in all-optical
networks [24]. It occurs due to power leakage from a new added channel 𝑬𝑎 (𝑡) of a similar
wavelength to that of the dropped channel 𝑬𝑑 (𝑡).
The receiver photo-current can be expressed as:
2
𝑖𝑝ℎ (𝑡) = 𝜌|𝑬𝑝ℎ (𝑡)| = 𝜌|𝑬𝑑 (𝑡) + 𝑬𝑎 (𝑡)|2 ( 2-6)
Equation (2-4) can be re-written as in [78]:
𝑖𝑝ℎ (𝑡) = [𝑃𝑑 + 𝑃𝑎 + 2√𝑃𝑑 𝑃𝑎 cos(𝜙𝑑 (𝑡) − 𝜙𝑎 (𝑡))] (2-7)
26
Equation (2-7) consists of three terms:
1. Dropped channel photo-current.
2. Added channel photo-current, which is small due to the crosstalk suppression.
3. Crosstalk current that results from the beating between 𝑬𝑎 (𝑡) and 𝑬𝑑 (𝑡). The worst
case is studied where the crosstalk term and the dropped signal are in phase [30].
Following the calculations of [67], the BER at the receiver is:
1 1 𝑖 − 𝐼𝐷 1 1 𝐼𝐷 (2-8)
𝐵𝐸𝑅 = 𝑒𝑟𝑓𝑐 ( ⋅ ) + 𝑒𝑟𝑓𝑐 ( ⋅ )
4 √2 𝛿1 4 √2 𝛿0
1. The first term of this equation represents the BER for a “1” state where the
receiver's current is 𝑖, while the second term is for a “0” state.
2. 𝐼𝐷 is the threshold level.
3. 𝑒𝑟𝑓𝑐 is the complementary error function [79].
4. 𝛿0 is the receiver noise which exists in the absence of crosstalk (it is mainly due to
thermal noise) for the “0” state.
5. 𝛿1 represents the sum of the beating term of crosstalk and receiver noise (𝛿1 =
√ 𝛿0 2 + 𝛾𝑖 2 ) for the “1” state [68].
𝑃
6. 𝛾 = 𝑃𝑎 is the crosstalk suppression ratio at resonance as shown in Figure 2-7.
𝑑
27
Figure 2-7. Crosstalk suppression of a single ring resonator OADM.
Using the optimum threshold value of 𝐼𝐷 given by [67]:
𝑖
𝐼𝐷 =
𝛿0 + √ 𝛿0 2 + 𝛾𝑖 2
BER is:
1 1
𝐵𝐸𝑅 = 𝑒𝑟𝑓𝑐 ( 𝑄)
2 √2
𝑖
where, 𝑄 = .
√ 𝛿𝑜 2 +𝛾𝑖 2
28
To evaluate the effect of intra-band crosstalk, power penalty should be considered. Power
penalty is defined as the amount of power to be added to overcome the effect of crosstalk
and maintain same BER in the absence of crosstalk.
The power penalty (𝑥) is [67]:
𝑥 = −10 log[1 − 𝛾 𝑄 2 ] (2-9)
For a BER=10−9, Q = 6 [73].
The power penalty required to counteract the intra-band crosstalk effect depends on 𝛾
(crosstalk suppression ratio) which in ring resonator based OADMs represents the
difference between the drop and through port responses at resonance (Figure 2-7). Equation
(2-9) is plotted in Figure 2-8 to show the relation between the crosstalk suppression ratio
(𝛾) and the required power penalty to obtain BER=10−9.
Figure 2-8. Power penalty as a function of crosstalk suppression ratio.
29
Crosstalk analysis shows that the level of crosstalk in ring resonator based OADMs is
lower than that in AWG based OADMs with less dependence on the number of channels in
the WDM signal. Also, ring resonator based OADMs are of small size and allow for high
scale integration compared to FBG based OADMs. Crosstalk mitigation techniques in ring
resonators were based on increasing the crosstalk suppression ratio as will be discussed in
the next section. In Figure 2-8, for the crosstalk suppression ratio being higher than |20|
dB, a high reduction in the imposed power penalty can be achieved. Allowing for high
values of crosstalk suppression ratios (at resonance) will reduce the required power penalty
for a narrow band of frequencies. However, for WDM networks, the crosstalk suppression
ratio should be kept high, ≥ |20| dB, over the whole frequency range of modulated
channels in order to ensure a reduced power penalty. Increasing the “crosstalk suppression
bandwidth” rather than the “crosstalk suppression ratio” allows adding/dropping modulated
channels in WDM networks with improved signal integrity and mitigated level of crosstalk,
and this is the main aim of this thesis.
2.5. Mitigation of Crosstalk in Ring Resonator Based
OADMs
The crosstalk effect in ring resonator based OADMs was studied numerically in [24, 25].
The intra-band crosstalk effects were estimated by calculating the eye opening penalty at
the receiver side. The drop port rejection ratio at a wavelength of the adjacent channel was
used to estimate the inter-band crosstalk effect. In [24], a numerical investigation of the
30
presence of input and add signals in a ring resonator OADM was performed. Modulated
signals at both the input and add ports were used to investigate the filter performance. The
analysis showed that, at high bit rate add/drop operations, the levels of induced intra-band
crosstalk and the wavelength selectivity are strongly dependant, even in higher order filters.
Multi-stage topology was suggested to reduce the effect of crosstalk for different data rates.
In this topology, the first stage is optimized for low crosstalk in the drop port channel,
while another stage is used for the added channel crosstalk mitigation. However, this
required an increase in the filter size.
The operation of OADMs based on active ring resonators was investigated in [25] with
high bit-rate return-to-zero (RZ) input channels at both the input and the add port. The use
of an active ring resonator was suggested in order to eliminate the intra-band crosstalk
between the incoming channels. A double-stage topology that addresses the inefficiencies
of the single stage OADM was proposed. However, the amplified spontaneous emission
(ASE) effect of the semiconductor optical amplifier was not considered.
In [23], lossy single ring and series coupled double ring OADMs were investigated
analytically and numerically in order to select appropriate coupling coefficients that reduce
the crosstalk at a given level of loss. Symmetric and asymmetric coupled ring resonators
were examined and the range of the appropriate coupling coefficients was estimated.
Limitations constraining the single ring OADM were addressed with the series coupled
double-ring OADM design. Vertical coupling was suggested as an alternative to the
31
standard coupled resonator layout, which allows greater flexibility in the choice of coupling
coefficients.
Series coupling between ring resonators was proposed to increase the filter order [26, 80].
Increasing the filter order leads to an improvement in the spectral response and allows high
suppression of the adjacent channel crosstalk (inter-band crosstalk). However, the sub-
micrometre gap between the rings (inter-ring coupling coefficient) has a great effect on the
overall response. In [26], the inter-ring coupling effect was addressed either by selecting
the optimum coupling or by proper physical arrangement of the rings. The optimum
condition for coupling coefficients to improve the crosstalk suppression ratio (only) was
proposed and a formula that calculates the optimum coupling coefficient of different order
filters was analytically derived. The optimum coupling coefficients for a second order
series coupled ring resonator in the presence of losses was studied (as will be discussed in
Chapter four).
In [80], the optimal arrangement for a high order series coupled ring resonator was
suggested. The dependence of the filter response of four series-coupled rings with two
different ring radii on the arrangement of ring radii was investigated. An analytical study to
calculate the effect of using rings with different radii on the inter-band crosstalk and how
these arrangements work with high bit rate signals was presented. However, this analysis
was applicable for the case of four rings resonators.
Parallel coupling between ring resonators was also proposed to mitigate the crosstalk [81,
82]. In this coupling configuration, the spectral response depends on the phase relationship
32
between rings (also controlled by the separation between rings). In [81], it was shown that a
number of ring resonators in parallel-coupled configurations provide an improvement in the
filter performance and reduce the inter-band crosstalk. Most attention was given to the out-
of-band rejection ratio (OBRR) and how to reduce the inter-band crosstalk by controlling
the separation between rings. In [82], the phase relationship between rings that affects the
spectral response of the filter was studied experimentally. A box-like response was
achieved, and high out of band rejection ratio was obtained. However, filter size was
increased.
A cross-grid architecture (using a vertical coupled ring resonator) was proposed to increase
the scalability. Cross-grid technology for crosstalk reduction was examined experimentally
in [10, 17]. In [10], the performance of a ring-resonator based OADM was evaluated
through the BER measurements in single channel 10 Gb/s and 3-channel 10 Gb/s WDM
configurations. The performance of three output ports with respect to a specified input one
was experimentally estimated. The robustness of this design was assessed with respect to
crosstalk effects when several channels propagate together. The drawback of this
technology is the intersection between optical waveguides that can lead to further crosstalk.
In [17], the role of cascaded OADMs for crosstalk reduction and spectrum clean-up in
add/drop filters were addressed experimentally for ring resonator cross-grid technology. An
add/drop node using a 2×2 cross-grid array and three ring resonators, to reduce the output
port crosstalk values, was proposed. In this design, the first ring was used to drop the
required channel, while the second and third rings were used to clean up the drop and
33
through port, respectively. The intersection between optical waveguides also represents the
drawback of this design.
A “Racetrack” model of the resonator was used in [83] to increase the through port notch
and improve the drop port response by using asymmetric coupling. In a racetrack resonator
based OADM, the coupling region length is longer than that of a ring resonator. This
permits better control of the spectral response. An increase in the crosstalk suppression
ratio was obtained. Increasing the filter order and also using a multi-stage structure were
proposed in [84], but at the expense of filter size.
2.6. Conclusion
In this chapter, crosstalk in all-optical networks was discussed and mathematically defined.
Optical add/drop multiplexers OADMs were taken as the main components of interest due
to their importance in accessing networks. Different structures of OADM were presented
and the crosstalk in each type was explained. An overview of the already existing
contributions to mitigate the crosstalk effect in ring resonator based OADMs were
presented. Increasing the “crosstalk suppression bandwidth” rather than the “crosstalk
suppression ratio” allows adding/dropping modulated channels in WDM networks with
improved signal integrity and mitigated level of crosstalk. Therefore, proposing solutions to
improve signal integrity by increasing the crosstalk suppression bandwidth in ring resonator
OADM is the main focus in the following chapters. However, an understanding of ring
resonators as a building block is required first, which is presented in Chapter three.
34
CHAPTER THREE
OPTICAL RING RESONATORS
In this chapter, Silicon-on-Insulator ring resonators are investigated and their add/drop
functionality is explained. Silicon waveguides and the coupling between evanescent modes
in directional couplers are discussed. Coupling schemes, cascaded coupling and coupling
coefficient effects on different port responses are discussed and explained. OADM spectral
characteristics are shown to be highly dependent on the coupling regions’ geometry. This
chapter provides the necessary background for the following chapters.
3.1. Introduction
Wavelength-Division Multiplexing (WDM) using Silicon-on-Insulator (SOI) waveguides
has become an attractive area of research to enable high integration density of photonic
components as well as to ensure high speed data transmission [16]. In SOI technology, high
index contrast between core and cladding materials allows for light propagation in small
cross-section silicon waveguides with very little optical leakage [7]. Therefore, SOI is
suitable for integrating photonic components in a micrometre transverse length scale [85].
35
CHAPTER THREE OPTICAL RING RESONATORS
WDM communication networks require optical components which can separate closely
spaced channels effectively and allow for the flexible addition and dropping of channels
[86]. Ring resonator based OADMs for WDM networks are considered as one example of
SOI technology [85]. Their small size allows for high density integration in optical
photonic circuits by exploiting the availability of the Complimentary-Metal-Oxide
Semiconductor (CMOS) fabrication facilities [19].
Ring resonators are promising devices for different applications in all-optical networks [87-
89]. Coupling a closed loop resonator with a bus waveguide leads to a modified structure
with a filter-like behaviour. Careful choice of coupling coefficients between ring and bus
waveguides has a great effect on the filter performance. Crosstalk analysis in Chapter two
suggested that ring resonator based OADMs have a superior performance over their FBG
and AWG counterparts. Therefore, this chapter aims to provide a general background of
optical ring resonators and present their features in WDM networks.
1. An explanation of ring resonators and their add/drop functionality in WDM networks is
presented. Directional couplers between ring and bus waveguides are discussed and
mathematically modelled.
2. The simulation software which is used throughout this thesis is introduced and
discussed.
36
3. Coupling schemes (vertical and lateral), cascaded coupling (series and parallel) and the
main criteria that define the usability of ring resonators as an OADMs, all are discussed
and explained.
3.2. Ring Resonators
Ring resonators were first proposed by Marcatili [90] to support travelling wave resonant
modes. A re-entrant waveguide with a perimeter of several µ𝑚 was used to construct an
optical resonator. The resonator was coupled to an external waveguide to get a transfer of
the optical energy. The resultant structure (shown in Figure 3-1) supports a number of
circulating wavelengths that satisfy the resonance condition 𝑁 ⋅ 𝜆𝑟𝑒𝑠 = 𝑛𝑒𝑓𝑓 ⋅ 𝑙, where 𝑁,
an integer representing the mode number, 𝑙 is the average resonator perimeter, and 𝑛𝑒𝑓𝑓 is
the effective refractive index. The difference between two consecutive resonances is called
the Free Spectral Range (FSR), which is of great interest in WDM systems.
If a WDM signal is launched at the input port in Figure 3-1, wavelengths that satisfy the
resonant condition will be coupled to the ring. The constructive interference after each
round trip results in an increase of the optical power in the resonator. The transfer of optical
power is realized by exploiting the coupling between the evanescent modes in the ring and
the bus waveguide. This structure represents a ring resonator based all-pass filter which is
used for dispersion compensation in WDM networks [91].
37
Figure 3-1. A schematic diagram of a ring resonator based all-pass filter.
In a ring resonator based OADM structure (Figure 2-5), there is another bus waveguide
coupled to the resonator. Therefore, the stored energy will be coupled to the output
waveguide leading to a build-up of optical power at the drop port and resulting in a notch in
the through port response (due to coupling) [92, 93]. The resonant wavelength is
determined by the resonator length and effective refractive index, whereas, the coupling
and loss coefficients are responsible for deciding the spectral response shape. Coupling
coefficients are dependent on the coupling region characteristics (separation gap and
coupling length), while losses are related to the type of materials used and the length of
resonator, as well [94]. Ring resonator based OADMs can be used to drop multiple
channels from a WDM signal to increase the flexibility of the network [95-97]. A four
channel dropping structure was proposed based on a compact parent-sub micro-ring
38
resonator [98]. Series and parallel coupled ring resonators have been proposed and used to
enhance the spectral characteristics of OADMs by increasing the Out-of-Band Rejection
Ratio and obtaining a sharp roll-off from pass-band to stop-band [21, 99-102].
Light propagation in any bounded medium is based on the refractive index contrast [30].
Low Index Contrast (LIC) materials were used first for optical waveguide fabrication where
the difference between core and cladding refractive indices is low. A reduction of the
propagation loss was achieved using the conventional LIC devices [103]. However, large
radius resonators were required to reduce leakage of the light. This means large
components with a small FSR (FSR is inversely related to the radius). The FSR is required
to be as high as possible to accommodate a wide bandwidth in the C-window (1535 −
1565 nm). Therefore, a number of rings (with different radii) were coupled in series to
increase the FSR using the Vernier effect [104-106]. The Vernier effect extends the FSR by
reducing all resonances which are not an integer multiple of the FSR of each individual
ring. The new FSR is related to the FSR of each ring as [107]:
𝐹𝑆𝑅𝑒𝑥𝑡𝑒𝑛𝑑𝑒𝑑 = 𝑛1 𝐹𝑆𝑅1 = 𝑛2 𝐹𝑆𝑅2
Where, 𝑛1 and 𝑛2 are integers.
The advancement in fabrication technologies has enabled the construction of small radii
resonators using high index contrast material (HIC) [108]. High index contrast between
core and cladding refractive indices results in a strong confinement of light even with a
small bend radius. Polycrystalline silicon (poly Si) waveguides were proposed in [109] with
radii of 3 , 4, and 5 𝜇𝑚 and a FSR about 20 − 30 nm. The disadvantage of this design was
39
the presence of a high insertion loss in the through port. Silicon nitride SiN was also used
in [84] and a ring of radius 8 𝜇𝑚 was fabricated to achieve 20 nm FSR. Recently, Silicon-
On-Insulator (SOI) has been used which allows for the fabrication of small radius rings
(using CMOS technology) with low bending and scattering losses [20]. A large FSR (up to
32 nm) with a low level of bending loss has been achieved [85].
Silicon-On-Insulator (SOI) waveguides are a promising technology for integrated photonic
devices in WDM networks [110]. In this technology the propagation loss is relatively low
[111]. However, the back reflection effect due to sidewall roughness is of great importance
[112, 113]. Sidewall roughness is usually considered as a random perturbation and back
reflection is treated as a stochastic variable [113]. The analytical calculations in [107]
showed that the performance of a resonator is strongly affected by the characteristics of the
surface roughness. The statistics of back reflection induced sidewall roughness were
investigated experimentally, first, in uncoupled optical waveguides [114]. It was shown that
the intensity of back reflection follows the distribution of a single scattering system with a
strong dependence upon waveguide length. Secondly, the change in back reflection
characteristics when a ring resonator is coupled to an optical waveguide was examined in
[115] and showed that after a multiple round trip in the ring resonator, back reflection
increases coherently and can affect the behaviour of the filter even at moderate quality
factors (high coupling coefficients). In rough-walled ring resonators, back reflection is a
well-known cause of resonance splitting due to mutual coupling between forward and
backward propagating modes [18, 107, 116-118]. This effect has been exploited to improve
40
the extinction ratio by increasing the depth of the through port at resonance [116] and is
further investigated in Chapter seven.
SOI ring resonators are receiving an increased level of attention from many research
groups. IMEC (Belgium) is one of the centres that work on the use of SOI single mode
optical waveguides [8, 18]. They fabricated a ring resonator with 5 µm radius [119] with
losses ranging from 2.5 − 3 dB/cm and FSR of 13.7 nm. Other groups such as the Institute
fur Halbeitertechni (Germany) [16, 120], California Institute of Technology [121], the
University of Wisconsin-Madison (USA) [122], and the Politecnico di Milano (Italy) [10,
115, 118], have fabricated ring and race track SOI resonators for different applications.
Reducing ring radius leads to an increase in the coupling coefficient sensitivity to
fabrication process (due to close proximity between ring and bus waveguides). To reduce
the coupling coefficient sensitivity, a straight waveguide section was introduced to increase
the coupling region. The resulting shape is a racetrack-like resonator (as shown in
Figure 2-5 b) [123]. However, this will increase the resonator perimeter and results in a
reduced FSR. Racetrack resonators with improved FSR were designed using the Vernier
effect [98, 124].
SOI ring resonators are a key building block to implement WDM schemes on CMOS
compatible platform and realizing monolithic integrated photonic circuits [19]. They have
found wide applications in all-optical networks such as add/drop multiplexers, delay lines,
and bio-sensors [87, 89, 118]. However, integration of optical components leads to
potential crosstalk and signal integrity (SI) issues due to the close proximity of optical
41
components and waveguides. In this thesis, a number of approaches are adopted to
effectively reduce the crosstalk between the channels in optical ring resonator based
add/drop multiplexers (OADMs) and filters in order to enhance the overall performance,
and, moreover, allow for increased integration.
3.3. SOI Strip Waveguides
Optical waveguides are the fundamental elements that interconnect different devices in
PLCs [125]. They have a similar function to that performed by metallic strips in an
electrical integrated circuit. However, unlike electrical signals that require a high
conductivity region to flow, optical signals require a high refractive index contrast medium
to propagate. SOI waveguides consist of a high refractive index material made of Si on the
top of lower refractive index SiO2 cladding layer on a silicon substrate [16], and fabricated
using UV or electron e-beam lithography technology [126]. Lithography is the process of
transferring patterns from mask to wafer [127].
Different optical modes travel in an optical waveguide with lateral and transverse
confinements. The optical mode is a spatial pattern of electromagnetic field in one or more
dimensions that remains constant in time. The number of transmitted optical modes
depends on the waveguide geometry and the choice of the materials [20]. To ensure single
TE mode propagation around 1.55 𝜇𝑚, the cross section of an SOI strip waveguide should
be between 200 𝑛𝑚 to 250 𝑛𝑚 in height and between 400 𝑛𝑚 to 500 𝑛𝑚 in width [18].
The most common way to couple light between optical waveguides is to place them close
42
to each other through a directional coupler. Coupling characteristics are affected by
geometrical dimensions of the coupling region, effective refractive index, and material
dispersion [125], as will be discussed in the following subsections.
3.3.1. Dispersion, Effective Refractive Index and Group Index
The resonance condition of ring resonators depends mainly on the waveguide effective
refractive index (𝑛𝑒𝑓𝑓 ). In order to perform a proper analytical modelling of coupling
coefficients, optical dispersion should be considered. Dispersion represents the rate of
change of the group delay with respect to wavelength. There are mainly two sources of
dispersion in optical waveguides:
1. Inter-modal dispersion: it is caused by the mixing of modes in a multi-mode system, and
it is of no concern under single-mode operation.
2. Chromatic dispersion: which includes:
i. Material dispersion, which comes from the wavelength-dependent index of a
material.
ii. Waveguide dispersion, which occurs when the speed of a wave or its eﬀective
index in a waveguide depends on its operating wavelength.
𝜕𝑛𝑒𝑓𝑓
The total dispersion ( ) of a guided mode is the sum of the material dispersion and
𝜕𝜆
waveguide dispersion.
43
The wavelength dependence of silicon and silicon dioxide refractive index (material
dispersion) has been determined in [128]. In both cases (Si and SiO2), the slope is negative
within the wavelength range of 1300 nm to 1700 nm.
𝑛(𝑆𝑖) = (−2.95 × 1016 )𝜆3 + (2.244 × 1011 )𝜆2 − (5.75 × 105 )𝜆 + 3.938 (3-1)
𝑛(𝑆𝑖𝑂2 ) = (−1.7 × 109 )𝜆2 − (7 × 103 )𝜆 + 1.459 (3-2)
The wavelength dependency of the effective refractive index for a single TE mode
waveguide shows a negative slope. Based on the curve-fitting results presented by [129],
the change of the effective refractive index is:
𝑛𝑒𝑓𝑓 (𝜆) = (−1.269 × 106 )𝜆 + 4.36 (3-3)
The group index is the partial derivative of the effective refractive index [124].
𝜕𝑛𝑒𝑓𝑓 (3-4)
𝑛𝑔 (𝜆) = 𝑛𝑒𝑓𝑓 (𝜆) − 𝜆
𝜕𝜆
Equations (3-3) and (3-4) represent the effective and group refractive indices dependence
on the wavelength which will be used in the mathematical analysis through this thesis.
3.3.2. Directional Coupler
Typically, directional waveguide couplers consist of two waveguides with close proximity
that permit a power exchange from one waveguide to the other [50, 125]. The length of
coupling region (𝐿c), distance between waveguides (g), and index profile of the coupler (n1,
n2 and n3) affect the amount of the coupled power. For the SOI directional coupler shown in
44
Figure 3-2, silicon dioxide (SiO2) of refractive index n2= 1.47 is used as a lower clad for
silicon (Si) waveguides with refractive index n1= 3.47; the upper cladding is air with
refractive index n3= 1.
Figure 3-2. Different geometries of SOI directional couplers.
The coupling coefficient 𝑘 is calculated based on the directional coupler geometry, as
below:
1. For a parallel bus waveguide directional coupler (Figure 3-2 a), the coupling of light can
be expressed in terms of the superposition of the supermodes [130]. The term “Supermode”
was used in [131, 132] to describe modes in a system of parallel coupled optical
waveguides. To ensure a 100% power transfer, the length of coupling region should be
equal to 𝐿𝜋 , where 𝐿𝜋 is the length over which the phase difference between odd and even
supermodes is equal to π.
The coupling region 𝐿𝜋 can be expressed as [16]:
45
𝜋 𝜆 (
𝐿𝜋 = = (3-5)
(𝛽𝑒 − 𝛽𝑜 ) 2(𝑛𝑒𝑓𝑓 𝑒 − 𝑛𝑒𝑓𝑓 𝑜 )
𝑛𝑒𝑓𝑓 𝑒 , and 𝑛𝑒𝑓𝑓 𝑜 are the effective refractive indices for even and odd supermodes,
respectively, which can be calculated using a numerical code, in [133, 134] (using the semi-
vectorial mode solver [135]).
Increasing the distance between waveguides (g) will reduce the difference (𝑛𝑒𝑓𝑓 𝑒 − 𝑛𝑒𝑓𝑓 𝑜 )
and leads to an increase in 𝐿𝜋 [83, 133]. The coupling coefficient 𝑘 depends on 𝐿𝜋 (which
is a function of g) and the actual length of coupling region LC (Figure 3-2) as in (3-6) [16]:
𝜋𝐿𝑐 (−(𝛽𝑒+𝛽𝑜 )𝐿 )
𝑐
(3-6)
𝑘 = sin ( )𝑒 2
2𝐿𝜋
2. For a directional coupler with two bent waveguides (Figure 3-2 b), (3-6) is not
applicable due to the change of the separation gap along the coupling region. The
instantaneous coupling strength is expressed in as (3-7) [50]:
𝜖𝑜 𝜔 (3-7)
𝑘(g(𝑧)) = ∬(𝑛1 2 − 𝑛3 2 )𝑓1 (𝑥, 𝑦)𝑓2 ∗ (𝑥, 𝑦)𝑑𝑥𝑑𝑦
4
Where, 𝑓1 (𝑥) and 𝑓2 ∗ (𝑥) are the power normalized modal fields in the two bent
waveguides, 𝜔 is the angular frequency and g(𝑧) is the separation width (centre to centre of
bent waveguides). The net coupling coefficient 𝑘 is:
∞
(3-8)
𝑘 = ∫ 𝑘(g(𝑧))𝑑𝑧
−∞
46
where, g(𝑧) depends on the waveguide width 𝑤𝑟 and the direction of propagation z. It can
be approximated to a parabola as in (3-9) [50]:
g(𝑧) = (g + 𝑤𝑟 ) + 𝛾𝑧 2 (3-9)
1 𝑅1 + 𝑅2
𝛾= =
2𝑅 2𝑅1 𝑅2
R1 and R2 are the radii of the bent waveguides.
3. For a bus and bent waveguide directional coupler (Figure 3-2 c). The coupling
coefficient can be calculated using (3-7) and (3-8) by substitute the radius 𝑅2 of the
bus waveguide as 𝑅2 → ∞.
Based on the above analysis, the coupling between the evanescent tails of modes in both
waveguides depends on the field profile which is difficult to estimate theoretically.
Therefore, 3D simulation software is used throughout this thesis for modelling and
estimating of coupling coefficients from the obtained frequency responses.
3.4. CST Microwave Studio
Analytical calculations, using Coupled Mode Theory, are not sufficient to provide the
required accuracy for OADM design. They do not account for multiple reflections very
easily, nor scattering and delay. Therefore, photonic CAD tools are needed for modelling
and validation. A wide variety of simulation algorithms (using Beam Propagation Method
(BPM), Eigenmode Expansion Method (EME), and Finite Difference Time Domain
Method (FDTD)) have been developed for modelling passive photonic devices [136-138].
47
In this thesis, CST Microwave Studio (MWS) [139] is used as a simulation engine to
simulate light propagation. It allows for both the validation of final designs and the analysis
through the design process (using the optimization options).
The full-wave discrete method used in CST MWS requires a mesh generation process to
specify the mesh cell size (by defining the Lines per Wavelength ratio (LPW)). Increasing
the LPW ratio will increase the accuracy of results, but at the expense of simulation time
and memory. A grid convergence test was carried out in this thesis to specify an acceptable
LPW ratio with a reasonable simulation time. The results showed that LPW= 15 to 20 is
within the acceptable range. Increasing the LPW ratio above 20 will result in a long
simulation time with little or no improvement in results. The CST simulation of the
fundamental mode profile is shown in Figure 3-3.
The CST MWS provides a simulation-driven optimization option which allows for
obtaining the optimized design parameters at a low computational cost. A proper
formulation of the objective function with a trust-region-based optimization algorithm
[140] will result in an optimum design for the OADM [141], as will be discussed in chapter
six.
48
Figure 3-3. CST simulations of the fundamental TE like mode distribution at the input port of a
single ring OADM
3.5. Coupling in Ring Resonators
There are two main approaches in which the coupling of light between the bent and bus
waveguides is achieved: lateral and vertical coupling. The waveguide cross section in each
scheme is different in order to support the required mode for each case. The TE like mode
is the dominant mode for the lateral coupling and for vertical coupling, the TM like mode is
dominant.
3.5.1. Lateral Coupling
If the bus and bent waveguides are placed in the same plane, as shown in Figure 3-4, the
coupling will take place horizontally. This is the lateral coupling configuration. The
49
coupling strength is controlled by the gap width between waveguides. The small gap size
required to ensure strong coupling in a HIC material makes this scheme of high sensitivity
to the lithography and etching procedures [127].
Figure 3-4. Laterally coupled ring resonator.
3.5.2. Vertical Coupling
In a vertical coupling configuration, bus and bent waveguides are etched in different layers
(as shown in Figure 3-5). From the design point of view, this means increased flexibility
because the ring and bus waveguides can be optimized separately [142]. The separation
layer thickness (d) and the lateral deviation between bus waveguides and the ring will affect
the coupling strength in this scheme [143]. Enhancing the crosstalk performance of a
vertical coupled OADM, by optimizing the ring parameters, will be the main aim of chapter
six.
50
Figure 3-5. Vertical coupled ring resonator.
3.6. Cascaded Coupled Ring Resonators.
Improved spectral characteristics such as flat pass band response, high out-of-band
rejection ratio and sharp step function can be obtained by using multiple (series or parallel)
coupled ring resonators.
3.6.1. Series Coupling
Figure 3-6 shows the schematic of series coupled double ring resonators. Several rings can
be placed between input and output bus waveguides. The outer coupling coefficients
(between bus waveguides and outer rings) and the coupling between inter-ring are modelled
51
by directional couplers with coupling coefficients, 𝜅𝑛 , and transmission coupling
coefficient 𝑡𝑛 . More explanation about the effect of inter-ring coupling is presented in
Chapter four.
If a defined WDM signal is injected as a source at the input port (port 1 in Figure 3-6), the
frequency response will be as follows:
1. Off-resonance, the fraction of light, which has completed a single round trip in the first
ring, interferes destructively with the light that has just coupled to the ring. There will be no
build-up of the power inside the resonator. Only a small amount of light will couple to the
second ring. The light remains mainly in the bus waveguides and propagates to the through
port.
2. At resonance, the fraction of light that has just completed one round-trip in the first ring
interferes constructively with the light coupled to the ring resulting in a coherent build-up
of the power inside the ring resonator. After multiple coupling between inner-rings, the
light will be dropped at port 3 (Figure 3-6) if the number of rings is odd, or it will be
dropped at port 4 (opposite to port 3) if the number of rings is even.
Ring radii are either of the same size to support similar resonance wavelengths or with
different sizes arranged to support a specific wavelength based on the Vernier effect [98,
124].
52
Figure 3-6. The schematic of N-series coupled ring resonator.
3.6.2. Parallel Coupling
In this configuration, rings are arranged in such a way that there is no direct coupling
between the nearest neighbouring rings (as shown in Figure 3-7). Therefore, it offers more
flexibility in the fabrication process compared to the serial configuration (no inter-ring
coupling). The centre-to-centre separation between the nearest neighbour rings (𝐿𝑝 ) will
determine filter response. Therefore, this distance should be chosen carefully to obtain the
desired interference at a specified wavelength range. The useful wavelength range is rather
limited due to the phase change that occurs due to the separation (𝐿𝑝 ) between the rings.
Outside this range the drop port response will vary in an undesirable way due to the
interference of light coming from the individual resonators. More explanation of the
53
separation 𝐿𝑝 effect on the crosstalk performance of this type of filters will be presented in
Chapter five.
Figure 3-7. The schematic of N-parallel coupled ring resonator.
3.7. Optical Add/Drop Multiplexer
In this configuration, two directional couplers are formed as shown in Figure 3-8.
Directional couplers are defined by the coupling coefficient (𝑘 2 ), and the transmission
coefficient ( 𝑡 2 ) [144]. The values of 𝑘 2 and 𝑡 2 are determined by the length of coupling
region, gap width and refractive index profile. For lossless coupling, 𝑘 2 + 𝑡 2 = 1.
54
However, the coupling losses are included in the loss coefficient 𝛼 of the ring which
determines the wave reduction after one round trip.
Figure 3-8 Schematic diagram of the ring resonator based OADM.
The spectral response of ring resonator based OADMs is highly affected by the coupling
region’s characteristics. Symmetrical coupled OADMs are realised by similar directional
couplers (i.e 𝑘1 = 𝑘2 ). OADMs are said to be asymmetrically coupled if 𝑘1 ≠ 𝑘2 which is
physically realized by a different separation gap in each side of the ring [102, 145]. The
drop and through port transfer functions of a single ring resonator OADM are calculated
using the space domain Coupled Mode Theory (CMT) [19, 94, 146], as in (3-10) and
(3-11), respectively:
−𝑘1 𝑘2 𝑥1/2 (3-10)

𝐺𝑑 =
1 − 𝑡1 𝑡2 𝑥
55
(𝑡1 − 𝑡2 𝑥) (3-11)
𝐺𝑡ℎ =
1 − 𝑡1 𝑡2 𝑥
where, 𝑥 = 𝑒 −𝛼𝑙−𝑗𝛽𝑙 is the round trip propagation coefficient and 𝑙 is the ring perimeter.
Coupling and loss coefficients affect all the filter parameters, starting from the insertion
loss to finesse as shown in equations (3-12) to (3-19) below [146]:
1. Through port response (Insertion loss):
𝑡2 2 𝑒 −2𝛼𝑙 − 2𝑡1 𝑡2 𝑒 −𝛼𝑙 𝑐𝑜𝑠𝛽𝑙 + 𝑡1 2 (3-12)

𝐼𝑅 = |𝐺𝑡ℎ |2 =
1 − 2𝑡1 𝑡2 𝑒 −𝛼𝑙 𝑐𝑜𝑠𝛽𝑙 + 𝑡1 2 𝑡2 2 𝑒 −2𝛼𝑙
At resonance (through port notch):
(𝑡2 𝑒 −𝛼𝑙 − 𝑡1 )2 (3-13)

𝐼𝑅𝑟𝑒𝑠 =
(1 − 𝑡1 𝑡2 𝑒 −𝛼𝑙 )2
The ratio between maximum and minimum values of the through port response represents
the extinction ratio, ER th:
2
𝐼𝑅𝑚𝑎𝑥 (1 − 𝑡1 𝑡2 𝑒 −𝛼𝑙 )(𝑡2 𝑒 −𝛼𝑙 + 𝑡1 )
𝐸𝑅𝑡ℎ = =[ ]
𝐼𝑅𝑟𝑒𝑠 (𝑡2 𝑒 −𝛼𝑙 − 𝑡1 )(1 + 𝑡1 𝑡2 𝑒 −𝛼𝑙 )
2. Drop port response:
𝑘1 2 𝑘2 2 𝑒 −𝛼𝑙 (3-14)
𝐷𝑅 = |𝐺𝑑 |2 =
1 − 2𝑡1 𝑡2 𝑒 −𝛼𝑙 𝑐𝑜𝑠𝛽𝑙 + 𝑡1 2 𝑡2 2 𝑒 −2𝛼𝑙
At resonance:
56
𝑘1 2 𝑘2 2 𝑒 −𝛼𝑙 (3-15)
𝐷𝑅𝑟𝑒𝑠 =
(1 − 𝑡1 𝑡2 𝑒 −𝛼𝑙 )2
The ratio between maximum and minimum values of the drop port response represents the
Out-of-Band Rejection Ratio, OBRR:
(1 + 𝑡1 𝑡2 𝑒 −𝛼𝑙 )2
𝑂𝐵𝑅𝑅 =
(1 − 𝑡1 𝑡2 𝑒 −𝛼𝑙 )2
3. Free spectral range (FSR): the frequency separation between two consecutive resonances
[18].
𝜆𝑟𝑒𝑠 2 (3-16)
𝐹𝑆𝑅 = 𝜆𝑟𝑒𝑠+1 − 𝜆𝑟𝑒𝑠 ≅
𝑛𝑔 𝑙
4. Full width at half maximum FWHM:
(1 − 𝑡1 𝑡2 𝑒 −𝛼𝑙 )𝜆𝑟𝑒𝑠 2 (3-17)

𝐹𝑊𝐻𝑀 =
𝜋𝑛𝑔 𝑙√𝑡1 𝑡2 𝑒 −𝛼𝑙
5. Q-factor, measures the number of round trips of the stored energy before it drops to
0.367 (= 1/𝑒) of its initial value; mathematically it can be expressed as:
𝜆𝑟𝑒𝑠 𝜋𝑛𝑔 𝑙√𝑡1 𝑡2 𝑒 −𝛼𝑙 (3-18)

𝑄𝑓𝑎𝑐𝑡𝑜𝑟 = =
𝐹𝑊𝐻𝑀 𝜆𝑟𝑒𝑠 (1 − 𝑡1 𝑡2 𝑒 −𝛼𝑙 )
6. Finesse: is a measure of the ratio between the resonances’ sharpness to their spacing;
mathematically, it can be expressed as [18]:
57
𝐹𝑆𝑅 𝜋√𝑡1 𝑡2 𝑒 −𝛼𝑙 (3-19)

𝐹𝑖𝑛𝑒𝑠𝑠𝑒 = =
𝐹𝑊𝐻𝑀 (1 − 𝑡1 𝑡2 𝑒 −𝛼𝑙 )
7. Crosstalk suppression: is defined as the difference between the drop and through port
intensities at the resonance. It represents the level of suppression of the unwanted
channels; mathematically it can be expressed as the difference between equations (3-12)
and (3-15).
8. Losses: the losses can be caused by different factors such as: coupling losses and round
trip losses (propagation losses, sidewall roughness and fabrication mismatch of
waveguide width). All the losses are included in the attenuation term (𝛼), and appear in
all equations as 𝑒 −𝛼𝑙 .
The key factors to assess the OADM performance are the through port notch depth,
maximum power dropped and the crosstalk suppression ratio at resonance. Equations (3-12)
to (3-19) show that a careful choice of the coupling coefficient will result in substantial
performance enhancement. A typical frequency response of a single ring resonator OADM
is presented in Figure 3-9. It was calculated using CST MWS. The through and drop port
responses are shown in this figure. Free spectral range (FSR) and Out-of-Band Rejection
Ratio (OBRR) are illustrated in this diagram.
58
Figure 3-9 CST simulated frequency response of a single ring resonator based OADM.
3.8. Conclusion
This chapter has presented an overview of the ring resonator operation principle,
fabrication and applications. Emphasis was given to their add/drop functionality in WDM
networks. Directional couplers in SOI waveguides were explained and the coupling
coefficients were mathematically defined. CST Microwave Studio (MWS) simulation
software was introduced and proposed to be used for numerical validation throughout this
thesis. Through and drop ports transfer functions were derived analytically using the space
domain CMT. The crosstalk suppression ratio (which is defined as the difference between
the drop and through port responses at resonance) was shown to be highly affected by
coupling coefficients. Controlling coupling coefficients through the waveguide cross
section, separation gap, and the length of coupling region allows for increased crosstalk
suppression ratio.
59
CHAPTER FOUR
OVER-COUPLED RING RESONATOR
BASED OADM
This chapter looks into crosstalk issues in (series) coupled ring resonator OADMs. It
examines the crosstalk performance as a function of the inter-ring coupling coefficients.
The over-coupling condition is proposed as a solution to improve the crosstalk suppression
bandwidth. The simulation results show that, in an over-coupled double ring resonator, the
crosstalk suppression bandwidth is increased in comparison with that of a critically
coupled OADM. Over-coupling improves filter performance by providing wider bandwidth
over which high crosstalk suppression is maintained.
4.1. Introduction
Series coupled ring resonators were proposed to enhance OADMs performance by
increasing the crosstalk suppression ratio and improving the spectral shape compared to
single ring based OADMs [124, 147]. Inter and intra-band crosstalk in series coupled
OADMs are highly affected by the inter-ring coupling coefficients. In [26] a formula to
calculate the optimum coupling coefficients that increase the crosstalk suppression ratio
was proposed, the resulting filter was said to be a critically coupled OADM. Inter-ring
60
CHAPTER FOUR OVER-COUPLED RING RESONATOR BASED OADM
coupling is responsible for the occurrence of a response splitting at resonance since the
coupling section will behave like a perturbation point in the ring [19]. Exploiting the
resonance splitting to increase the crosstalk suppression bandwidth is the main aim of this
chapter.
In an ideal resonator (smooth-walled ring and without a coupling section) each mode can
travel in two directions, the forward propagating mode (deliberately excited by the bus
waveguide) and the backward propagating mode; the forward and backward travelling
modes are uncoupled [18]. However, any small perturbation which can be felt by the
optical mode can lead to couple these two modes. When there is coupling, this will result in
a net power transfer between modes, and causes resonance splitting.
The outer coupling coefficients (between rings and bus waveguides) and inter-ring coupling
coefficients play an important role to control the resonance splitting. Figure 4-1 shows the
effect of inter-ring coupling coefficient in a series double ring resonator. The outer coupling
coefficient 𝑘 2 is chosen to be 0.24. Different values of the inter-ring coupling coefficients
𝑘𝑖 2 are examined (𝑘𝑖 2 = 0.016, 0.05, 0.1, 𝑎𝑛𝑑 0.15) to show how the resonance splitting is
changed with the increase of inter-ring coupling. For 0.016 inter-ring coupling, the drop
port response shows no splitting. However, for a 0.15 coupling coefficient (decrease of the
gap separation between rings), the splitting becomes strong with a single minimum at
resonance of – 5dB. At the same time, the through port response will also have resonance
splitting and a single maximum will appear at resonance. Physically, that can be interpreted
as: if the time to deplete the ring due to counter directional coupling (depending on the back
61
reflection coefficient) becomes shorter than the time to charge up the ring, the resonance
splitting will exist [50].
Figure 4-1. Analytically calculated drop port response of a series coupled RR, with a study of
the inter-ring coupling (𝒌𝒊 𝟐 ) effect on the resonance splitting.
Keeping the difference between this single minimum at the drop port and the single
maximum of the through port (at resonance) exceeds an adequate level of suppression for a
wide range of wavelengths allows for high crosstalk suppression bandwidth and improves
filter performance. This chapter aims to:
1. Derive an expression for the crosstalk suppression bandwidth in single and double
ring resonators OADM.
2. Maximize the crosstalk suppression bandwidth by exploiting the resonance splitting
induced by the inter-ring coupling.
62
1. Analytical models of the crosstalk suppression bandwidth in a single and double ring
resonator are derived and validated using CST MWS.
2. The inter-ring coupling coefficient effect on the crosstalk suppression bandwidth is
modelled and optimized.
3. An over-coupling condition between inter-ring is proposed and simulated.
This chapter concludes with the design of an over-coupled OADM that provides 40 GHz
crosstalk suppression bandwidth compared to 28 GHz in the critically coupled OADM.
Increasing the bandwidth means that higher data rate channels can be added/dropped with
improved signal integrity.
4.2. Crosstalk Bandwidth in a Single Ring Resonator
Figure 4-2 shows a schematic diagram of a single ring resonator based OADM. It indicates
that a single ring resonator is coupled with two bus waveguides via two coupling regions.
In this section, the crosstalk performance of a single ring resonator based OADM is
analysed analytically and modelled using CST MWS to estimate the crosstalk suppression
bandwidth.
63
Figure 4-2. Schematic of a single ring add/drop filter with three NRZ of 10 Gbps modulated WDM
signal.
4.2.1. CST Simulation
The time domain solver results using CST MWS [139] for the single ring resonator
(illustrated in Figure 4-2) are shown in Figure 4-3. An SOI ring resonator was modelled
using the following parameters: ring radius equal to 16 μm (corresponding to a 5.5 nm free
spectral range), a silicon waveguide with a core refractive index (𝑛𝑐𝑜𝑟𝑒 = 3.47), and a
silicon dioxide lower cladding with a core refractive index (𝑛𝑐𝑙𝑎𝑑 = 1.47). The upper clad
refractive index was equal to 1 (air). The cross section of silicon waveguide was chosen to
ensure a single mode propagation (width= 460 nm × height= 250 nm). In Figure 4-3, 𝑆21
and 𝑆31 represent through port and drop port frequency responses, respectively, while, 𝑆11
and 𝑆41 are the normalized power reflection at input port and add port respectively. The
64
separation between bus and ring waveguides [g] was 40 nm which corresponds to a power
coupling coefficient (𝑘1 2 = 𝑘2 2 ≈ 0.4). For this value of coupling, the through port
attenuation at resonance is −20.7 dB and the maximum drop port is about −1.05 dB,
which means that it is difficult with this coupling value to maintain a useful bandwidth.
Different coupling coefficients are examined by changing the separation between the bent
and straight waveguides to maximize the crosstalk suppression bandwidth.
Figure 4-3. Frequency response of a single ring resonator.
4.2.2. Analytical Calculations
The inter-band crosstalk level is obtained from (3-14) by calculating 𝐷𝑅 at 100 GHz,
which is called the ‘Drop Port Rejection Ratio’ DPRR. However, this value depends on the
65
data rate. For example, it should be calculated at 200 GHz when the transmission rate is 40
Gbps since the channel separation is 200 GHz. The intra-band crosstalk is calculated by
taking the difference between the through port intensity and drop port intensity at resonance
and it should be high to ensure high crosstalk suppression ratio. To obtain a high
suppression of crosstalk in a single ring resonator, the through port intensity should vanish
at resonance to ensure that the required channel is completely dropped. This implies that
(3-12) should be equal to zero. In a symmetric ring resonator it is difficult to get the
numerator of (3-12) equal to zero except in the case of a lossless resonator which is simply
impractical. An asymmetric ring resonator with different coupling coefficients was used to
maximize crosstalk suppression by choosing 𝑡1 and 𝑡2 in such a way that (3-12) equals
zero. These values of coupling coefficients represent “critical” coupling [18].
The other important factor is to calculate the bandwidth over which the crosstalk
suppression value is more than the minimum acceptable level of |20| dB [23]. By taking
the difference between (3-14) and (3-12), the crosstalk suppression 𝑋𝑇 is given as:
𝑋𝑇 = 10 log 𝐷𝑅 − 10 log 𝐼𝑅 (4-1)
𝑘1 2 𝑘2 2 𝑒 −𝛼𝐿 (4-2)
𝑋𝑇 = 10log
1 + 𝑡1 2 𝑡2 2 𝑒 −2𝛼𝐿 − 2𝑡1 𝑡2 𝑒 −𝛼𝐿 𝑐𝑜𝑠 𝛽𝐿
𝑡1 2 + 𝑡2 2 𝑒 −2𝛼𝐿 − 2𝑡1 𝑡2 𝑒 −𝛼𝐿 𝑐𝑜𝑠 𝛽𝐿

− 10log
1 + 𝑡1 2 𝑡2 2 𝑒 −2𝛼𝐿 − 2𝑡1 𝑡2 𝑒 −𝛼𝐿 𝑐𝑜𝑠 𝛽𝐿
The shape of XT suppression is symmetric around the resonance frequency and the
wavelength shift from resonance for each value of XT is calculated using (4-3):
66
𝑡1 2 + 𝑡2 2 𝑒 −2𝛼𝐿 − 𝑘1 2 𝑘2 2 𝑒 −𝛼𝐿 × 10−𝑋𝑇/10 (4-3)

Cos 𝛽𝐿 =
2𝑡1 𝑡2 𝑒 −𝛼𝐿
For example, for the required value of |20| dB, the bandwidth of crosstalk suppression is
found from:
𝑐 𝑡1 2 + 𝑡2 2 𝑒 −2𝛼𝐿 − 0.01 × 𝑘1 2 𝑘2 2 𝑒 −𝛼𝐿 (4-4)

∆𝑓 = [2𝜋𝑁 + cos −1 ]
𝜋𝑛𝑒𝑓𝑓 𝐿 2𝑡1 𝑡2 𝑒 −𝛼𝐿
where, N is the mode index of the resonator, and c is the speed of light in free space.
Figure 4-4 shows the crosstalk bandwidth and the DPRR as a function of coupling
coefficients for a symmetric single ring resonator. Changing coupling coefficients will
affect filter selectivity through the change of quality factor (3-18).
Figure 4-4. Bandwidth of crosstalk suppression and DPRR for a single ring resonator.
The bandwidth of modulated channels is mainly affected by the data rate and the
transmission technique. For a given data rate, the bandwidth of RZ is twice that of NRZ. 10
67
Gbps of NRZ transmission requires 20 GHz bandwidth. Thus, the crosstalk suppression
bandwidth should be more than 20 GHz to ensure a dropping of this channel with a reduced
level of crosstalk. For a symmetric ring resonator, and from (4-4), the value of the coupling
coefficient to attain the required bandwidth for 10 Gbps NRZ is |𝑘|2 = 0.625. For this high
coupling coefficient, the DPRR at 100 GHz can be calculated from (3-14) and it is found to
be equal to 2.4 dB, which is very low and means that a high level of inter-band crosstalk
will be added to the dropped signal. If a 10 Gbps RZ signal is used, the single ring
resonator filter is unable to support this signal since it requires a high coupling coefficient
(about 0.8) which gives a 1.8 dB DPRR (high inter-band crosstalk).
Asymmetric coupling may represent a better alternative where the coupling coefficient of
the second coupling region (𝑡2 ), can be chosen to satisfy the critical coupling condition
𝑡1 = 𝑡2 𝑒 −𝛼𝐿 . Using these coupling coefficients for both coupling regions, the through port
response at resonance is equal to zero (notch filter). This gives maximum crosstalk
suppression ratio. The limitation of the critical coupling ring resonator is that it is
asymmetric, which means that the signal entering from the input port is subjected to a
different coupling value from the signal entering the adding port. This result in a different
behaviour for the add and drop ports of the filter. This limitation has been addressed in [81]
by suggesting a series-cascaded ring resonator pair in which the coupling coefficients are
adjusted to reduce the crosstalk in the added and dropped signals simultaneously. The
effect of coupling coefficients in an asymmetric ring resonator is shown in Figure 4-5.
68
Figure 4-5. Bandwidth for asymmetric single ring resonator as a function of coupling coefficients.
Figure 4-6. Bandwidth of a symmetric single ring resonator as a function of coupling coefficient and
losses.
The surface plot presented in Figure 4-5 shows that an asymmetric single ring resonator
requires a high coupling coefficient to ensure a wide bandwidth. High through port
attenuation is possible in a critical coupled single resonator. However, the bandwidth over
69
which this suppression occurs is narrow. To increase this bandwidth the coupling
coefficient should be high, which would act to increase the level of inter-band crosstalk.
The presence of losses in ring resonators also affects the useful bandwidth, as shown in
Figure 4-6. Increasing losses in the ring will reduce the bandwidth of crosstalk suppression,
resulting in a high inter- and intra-band crosstalk.
4.3. Crosstalk Bandwidth in Double Ring Resonator
An improved spectral response can be obtained when multiple ring resonators are coupled
in series. A high DPRR will result from increasing the filter order and this in turn will
improve the inter-band crosstalk suppression. However, the study of intra-band crosstalk in
second order ring resonators shows a high dependence on the coupling coefficients between
the rings (inter-ring coupling 𝑘𝑖 ). The outer coupling coefficients (between bus waveguides
and rings) are considered symmetric (identical separation of outer coupling regions).
Hence, the crosstalk performance will depend on the choice of the inter-ring coupling value
relative to the outer coupling value. The calculations of the optimum values of inter-ring
coupling coefficient 𝑘𝑖 for different orders of series coupled ring resonator were shown to
𝑘2
follow the formula [ 𝑘𝑖 = (2−𝑘 2 ) ] [26], where 𝑘 is the coupling coefficient of the outer
rings. It was shown in [25] that the optimum value of coupling coefficient calculated using
the above formula yields a higher through port attenuation at resonance and maximizes
crosstalk suppression ratio. Table 4-1 shows the values of inter-ring coupling coefficients
corresponding to the outer coupling coefficients for a symmetric double ring resonator.
70
Table 4-1. The relation between the inner and outer coupling coefficients for optimum coupling.
Outer-coupling coefficient
𝟎. 𝟓 𝟎. 𝟓𝟓 𝟎. 𝟔 𝟎. 𝟔𝟓 𝟎. 𝟕 𝟎. 𝟕𝟓 𝟎. 𝟖
𝟐
[|𝒌| ]
Inter-coupling coefficient
𝟎. 𝟏𝟏𝟑 𝟎. 𝟏𝟒𝟓 𝟎. 𝟏𝟖 𝟎. 𝟐𝟑 𝟎. 𝟐𝟗 𝟎. 𝟑𝟔 𝟎. 𝟒𝟒𝟒
[|𝒌𝟐 |𝟐 ]
The optimum values in Table 4-1 maximize crosstalk suppression ratio, but do not give the
maximum bandwidth (required to ensure a high bit rate transmission in the network). An
analysis of the spectral response of a higher order ring resonator shows that the response
exhibits a “splitting” at resonance, which depends on the inter-ring coupling coefficient,
and the number of splits is dependent upon the order of the filter [18].
The analysis given in this section focusses on the calculation of crosstalk suppression
bandwidth in a second order ring resonator and the values of coupling coefficient that
maximize it. The aim is to find the value of the inter-ring coupling coefficient that produces
a splitting of the response at resonance with a level that satisfies the crosstalk suppression
requirements over as wide bandwidth as possible. The drop port response will have a
double maximum near the resonant frequency and a single minimum at resonance. The
through port will have a single maximum at resonance and a double minimum near
resonance. By keeping the difference between the single minimum of the drop port
response and the single maximum of the through port within the accepted level of
suppression, the bandwidth will be increased. To operate the filter in a “splitting” spectral
71
region the inner coupling coefficient should be higher than the optimum coupling values
given in Table 4-1, and hence these are described as “over-coupled” ring resonators.
Figure 4-7. The schematic of a series double ring resonator add/drop filter with 10 Gbps RZ WDM signal.
Following the analysis given in section 4.3, for the second order ring resonator shown in
Figure 4-7, the drop port and through port transfer functions and the intensity responses are
given as below [23]:
The drop and through ports transfer functions are:
𝑡1 − 𝑡2 𝑥 − 𝑡1 2 𝑡2 𝑥 + 𝑡1 𝑥 2 (4-5)
𝑇𝑡ℎ𝑟𝑜𝑢𝑔ℎ =
1 − 2𝑡1 𝑡2 𝑥 − 𝑡1 2 𝑥 2
72
𝑗𝑘1 2 𝑘2 𝑥 2 (4-6)
𝑇𝑑𝑟𝑜𝑝 =
1 − 2𝑡1 𝑡2 𝑥 − 𝑡1 2 𝑥 2
The intensity responses for drop and through ports are:
𝐴1 − 𝐴2 cos 𝛽𝐿 + 𝐴3 cos 2𝛽𝐿 (4-7)

𝐼𝑅 =
𝐵1 − 𝐵2 cos 𝛽𝐿 + 𝐵3 cos 2𝛽𝐿
𝐷 (4-8)
𝐷𝑅 =
𝐵1 − 𝐵2 𝑐𝑜𝑠 𝛽𝐿 + 𝐵3 𝑐𝑜𝑠 2𝛽𝐿
where,
𝐴1 = 𝑡1 2 + 2𝑒 −2𝛼𝐿 𝑡1 2 𝑡2 2 + 𝑒 −2𝛼𝐿 𝑡2 2 + 𝑒 −2𝛼𝐿 𝑡1 4 𝑡2 2 + 𝑒 −4𝛼𝐿 𝑡1 2 (4-9.a)
𝐴2 = 2𝑒 −𝛼𝐿 𝑡1 𝑡2 + 2𝑒 −3𝛼𝐿 𝑡1 𝑡2 + 2𝑒 −𝛼𝐿 𝑡1 3 𝑡2 + 2𝑒 −3𝛼𝐿 𝑡1 3 𝑡2 (4-9.b)
𝐴3 = 2𝑒 −2𝛼𝐿 𝑡1 2 (4-9.c)
𝐵1 = 1 + 4𝑒 −2𝛼𝐿 𝑡1 2 𝑡2 2 + 𝑒 −4𝛼𝐿 𝑡1 4 (4-9.d)
𝐵2 = 4𝑒 −𝛼𝐿 𝑡1 𝑡2 + 4𝑒 −3𝛼𝐿 𝑡1 3 𝑡2 (4-9.e)
73
𝐵3 = 𝐴3 (4-9.f)
𝐷 = 𝑒 −2𝛼𝐿 [1 − 𝑡2 2 − 2𝑡1 2 + 2𝑡1 2 𝑡2 2 + 𝑡1 4 − 𝑡1 4 𝑡2 2 ] (4-9.g)
Figure 4-8.a shows the frequency response of a critical coupled double ring resonator
obtained using the time domain solver modelled in CST MWS. S41 represents the drop port
frequency response in the double ring resonator. The coupling coefficient was (≈ 0.4) and
the inter-ring separation was 90 nm. The crosstalk bandwidth of about 28 GHz is shown in
this Figure. However, Figure 4-8.b is the response of the same filter with a different
separation between the rings. Changing the inter-ring separation to 85 nm will lead to the
occurrence of frequency splitting at resonance. This splitting comes from the mutual
coupling between the forward and backward modes propagated inside the ring (as discussed
in section 4.1). However, this process could be interpreted as a useful effect to increase the
bandwidth of crosstalk suppression as shown in Figure 4-8.b. Figure 4-8.b shows an
increase in the useful bandwidth over which higher crosstalk suppression is obtained.
74
(a)
(b)
Figure 4-8. a) Frequency response for series double ring resonator. b) Spectrum splitting at
resonance for series double ring resonator.
75
Analytically, using the same approach used in the case of a single ring resonator, the
crosstalk suppression bandwidth is calculated as in (4-1). The difference here is the
presence of a single maximum and minimum in the through and drop response respectively.
To examine the effect of changing the inter-ring coupling coefficient on the spectral
response, the second derivative of the drop port response given in (4-8) with respect to the
phase will be taken. At resonance, the value of 𝑘𝑖 that gives the second derivative a value
equal to zero (maximally flat) is calculated and designated as 𝑘𝑓𝑙𝑎𝑡 . To maximize the
bandwidth, the value of inter-ring coupling is chosen to be greater than 𝑘𝑓𝑙𝑎𝑡 and should
satisfy (4-1) where the value of 𝑋𝑇 should be held at |20| dB or more.
From (4-1) and using (4-7) and (4-8), (4-10) is obtained
𝐷 (4-10)
𝑋𝑇 = 10 log
𝐴1 − 𝐴2 𝑐𝑜𝑠 𝛽𝐿 + 𝐴3 𝑐𝑜𝑠 2𝛽𝐿
For the required level of crosstalk suppression:
𝐴1 − 𝐴2 𝑐𝑜𝑠 𝛽𝐿 + 𝐴3 𝑐𝑜𝑠 2𝛽𝐿 = 0.01𝐷 (4-11)
In order to calculate the bandwidth of crosstalk suppression from (4-11), this equation
should first be solved to obtain the value of inter-ring coupling that ensures the required
level of suppression at resonance frequency (where the single maximum and minimum of
through and drop response occurs). At resonance, (4-11) results in a second order equation
for 𝑡2 2 and the solution of this equation will result in two values of 𝑡2 . The value of inter-
ring coupling (𝑘2 = √1 − 𝑡2 2 ) that results in an over-coupling between rings should be
greater than 𝑘𝑓𝑙𝑎𝑡 as in Table 4-2.
76
Table 4-2. Inter-ring coupling coefficients for over-coupling.
Outer-coupling coefficient
𝟎. 𝟓 𝟎. 𝟓𝟓 𝟎. 𝟔 𝟎. 𝟔𝟓 𝟎. 𝟕 𝟎. 𝟕𝟓 𝟎. 𝟖
𝟐
[|𝒌| ]
Inter-coupling coefficient
𝟎. 𝟏𝟑𝟏 𝟎. 𝟏𝟕𝟎 𝟎. 𝟐𝟏𝟕 𝟎. 𝟐𝟕𝟑 𝟎. 𝟑𝟒 𝟎. 𝟒𝟏 𝟎. 𝟓𝟏𝟐
|𝟐
[|𝒌𝟐 ]
Equation (4-11) should be solved again as a second order equation in terms of 𝑐𝑜𝑠 𝛽𝐿 in
order to calculate the bandwidth. Using (4-12), the bandwidth can be calculated in terms of
coupling coefficients
(4-12)
𝐴2 𝐴2 2 (𝐴1 − 0.01𝐷)
cos 𝛽𝐿 = + 0.5√[( ) − 4 ]
2𝐴3 𝐴3 𝐴3
The values for 𝐴1 , 𝐴2 , 𝐴3 and 𝐷 are calculated from (4-9.a).
Figure 4-9 shows the difference in the crosstalk suppression bandwidth for critical coupling
(Table 4-1) and that computed using (4-12). It is noted that for RZ at 10 Gbps, where the
required bandwidth is 40 GHz, the power coupling coefficient in the case of critical
coupling is 0.58, while in the case of an over-coupled ring resonator it is 0.46. The DPRR
in the case of an over coupled ring resonator is 13 dB. This is compared to 7 dB for critical
coupling which means an enhancement of inter-band crosstalk suppression.
77
Figure 4-9. Bandwidth of crosstalk suppression and drop port rejection ratio for critical and over
coupled double ring resonator (losses =4 dB/cm).
The effect of ring losses in the over-coupled ring resonators is shown in Figure 4-10.
Different values of round trip losses are used, and for each value of loss the inter-ring
coupling coefficient is calculated to obtain the required level of crosstalk. Then the
bandwidth of crosstalk suppression is calculated as a function of both the losses and
coupling coefficients. For some levels of losses it is difficult to acquire the accepted level
of crosstalk suppression. This is represented as a bandwidth equal to zero in Figure 4-10.
From the results shown in Figure 4-9 and Figure 4-10 it can be seen that an over-coupled
ring resonator provides a wider bandwidth to accommodate higher data rate signals with
low crosstalk compared to a critical coupled ring resonator filters. Optimising the value of
inter-ring coupling to operate in the over coupled region gives a better spectral response in
terms of intra-band and inter-band crosstalk simultaneously.
78
Figure 4-10. Bandwidth of the over coupled ring resonator as a function of losses for different
coupling coefficient.
4.4. Conclusion
In this chapter, increasing the crosstalk suppression bandwidth in a series (double) ring
resonator based OADM was the main aim of interest. Inter and intra-band crosstalk effects
in single and double ring resonator filters were investigated for intensity modulated RZ and
NRZ signals. It was shown that for a double ring resonator based filter, with a power
coupling coefficient of 0.46, the bandwidth of crosstalk suppression in the case of critical
coupling is 28 GHz, while for the over coupled condition the bandwidth is 40 GHz. This
means that a critically coupled filter will add more crosstalk if used with a 10 Gbps RZ
signal. Physically, that means, for the same separation gap between the rings and bus
waveguides, it is possible to enhance the bandwidth of the model by more than 40% by
79
adjusting the inter-ring coupling. Over coupling in a series coupled OADM improves the
bandwidth and the level of inter-band crosstalk simultaneously and allows for high data rate
channel dropping.
80
CHAPTER FIVE
CROSSTALK BANDWIDTH ESTIMATION
OF PARALLEL COUPLED OADM
This chapter investigates crosstalk issues in (parallel) coupled optical ring resonators. It
examines the performance of a well-known optical device (an OADM based on parallel
coupled ring resonators realized in SOI technology) but focusses on the optical signal
integrity perspective. The Signal Flow Graph approach (based on Mason’s rule) is used to
identify filter performance in terms of crosstalk suppression bandwidth and EMC. The use
of Mason’s rule in this regard is novel and useful. The good agreement between analytical
and simulation results suggests that using the derived analytical model is a faster and
easier approach for filter design and provides a better insight into the signal integrity
performance of the filter.
5.1. Introduction
Parallel coupled ring resonators were proposed to enhance the overall response of OADM
in WDM networks [81, 82, 102]. High Out-of-Band Rejection Ratio (OBRR) and improved
crosstalk suppression ratio are shown to be achieved by optimizing coupling coefficients
and the separation distance between rings. However, optimizing the resonator parameters,
81
CHAPTER FIVE CROSSTALK BANDWIDTH ESTIMATION OF PARALLEL COUPLED OADMs
in order to enhance the crosstalk performance, requires a simple and direct form for the
OADM transfer function.
Different techniques have been proposed for the analysis of cascaded photonic devices. The
most elementary analytical method is to write out explicit node and loop equations and
extract the overall transfer function from them. Despite the fact that this method provides
the required characteristics (such as phase, group delay and dispersion) it is complicated
and cumbersome [20]. The transfer matrix based method has been used for transfer function
derivation by calculating the scattering matrix of each ring. The overall transfer matrix is
calculated by using matrix multiplication [148]. The complexity of this approach increases
with the number of coupled rings.
The graphical approach, also called Signal Flow Graph (SFG) method proposed by Mason
[149] was also used to provide a faster and easier approach for multipath (series coupled)
ring resonators [124]. This approach showed a reduction of calculation time from 1/3 to
1/20 compared to the transfer matrix method, depending on the complexity of the filter
(number of rings) [150]. The group delay and dispersion characteristics of the filter are
difficult to calculate directly by the SFG method. However, in [151] this method was used
to calculate the transfer function of a single ring resonator OADM including the group
delay. This chapter aims to:
1. Derive the transfer function of a parallel coupled ring resonator OADM using the
SFG method.
82
2. Optimize filter parameters (based on the derived transfer function) in order to
improve the crosstalk suppression bandwidth.
1. The SFG method based on Mason’s rule is presented and a general form of the transfer
function of N rings coupled in parallel is derived.
2. The validity of the proposed analytical model is examined against CST MWS simulation
results.
3. Coupling coefficients, centre-to-centre separation and ring losses effects are studied and
optimized to improve the crosstalk suppression bandwidth.
This chapter concludes with a simple and direct form for the transfer function. This form is
used to study the effect of different parameters on the filter performance and, moreover, to
improve the crosstalk suppression bandwidth.
5.2. Mason’s Rule for Parallel Coupled Ring Resonators
Consider the parallel coupled ring resonator based filter shown in Figure 5-1. If a WDM
signal enters at port 1, only the channels that satisfy the resonance condition will be
coupled to port 3, the designated drop port. Channels with different wavelengths from
resonance will pass (ideally) unaffected to the through port. A new channel can be added at
port 4 (the opposite side of the system to port 3).
83
Figure 5-1. CST model of parallel coupled ring resonator.
The coupled light in each ring will be subjected to a transmission coefficient x1 given in
(5-1) after one round trip, while the light propagating from the first ring to the second ring
will have a magnitude and phase change depending on x2 given in (5-2), which represents
the waveguide loss and propagation phase change.
x1 = 𝑒 −𝛼𝐿𝑟 −𝑗𝛽𝐿𝑟 (5-1)
x2 = e−αLp−jβLp (5-2)
In (5-1) and (5-2), 𝛽 is the phase constant, 𝐿𝑟 is the ring perimeter, 𝐿𝑝 is the separation
distance between the rings and 𝛼 is the loss coefficient measured in cm−1.
84
There are a number of assumptions on which this analysis is based:
1. Losses in the bus waveguides are small, so the approximation can be made as
e−αLp ≈ 1.
2. Propagation constants in the ring and bus waveguides are the same.
3. No back reflection propagation occurs in the rings.
4. Equal radii rings, 𝐿𝑟 = 𝐿𝑟 1 = 𝐿𝑟 2 .
5. Symmetric coupled rings are used.
Mason’s rule [149] was used to determine the transfer function of a linear system by first
finding the forward paths between input and output and then defining the closed loop gains.
The total gain is then calculated by taking the summation of the forward gain of each path
multiplied by a quantity representing the gain of non-touching loops of that path. This
quantity is called the cofactor. The result is divided by a term representing the total gain of
closed loops plus the gain product of non-touching loops. The two loops are said to be non-
touching if they do not share any node.
In Figure 5-2, there are two forward paths: G1 represents the path for light coupled to the
first ring, propagating around the half length of this ring and coupled to the drop port. The
remaining light will propagate in the second path G2 , which is twice the distance between
rings plus half of the second ring length.
85
Figure 5-2. Forward paths.
G1 = −𝑘1 2 x11/2 (5-3)
G2 = −t1 2 𝑘2 2 x2 2 x11/2 (5-4)
There are two closed loops as shown in Figure 5-3 resulting from the light circulating in
each ring (T1 and T2 ), and extra closed loop resulting from the path containing half of each
ring and the separation between them (T3 ).
86
Figure 5-3. Closed loops.
T1 = t1 2 x1 (5-5.a)
T2 = t 2 2 x1 (5-5.b)
T3 = 𝑘1 2 𝑘2 2 x1 x2 2 (5-5.c)
The gain product of non-touching loops (T1 and T2 ) is given by:
T21 = t1 2 t 2 2 x1 2 (5-6)
87
Applying Mason’s rule:
∑ Gi ∆i (5-7)
G=
∆
where, G and Gi are the total gain of the system and the gain of each forward path,
respectively. ∆ is the determinant of the closed loops, and ∆i is the cofactor for each path.
∆= 1 − t1 2 x1 − t 2 2 x1 − 𝑘1 2 𝑘2 2 x1 x2 2 + t1 2 t 2 2 x1 2 (5-8.a)
∆1 = 1 − T2 − T3 (5-8.b)
= 1 − t 2 2 x1 − 𝑘1 2 𝑘2 2 x1 x2 2
∆2 = 1 − T1 = 1 − t1 2 x1 (5-8.c)
The gain at the drop port is:
G1 ∆1 + G2 ∆2
G=
∆
1 3 3 1
−𝑘1 2 x1 2 +t2 2 𝑘1 2 x1 2 +𝑘1 4 𝑘2 2 x1 2 x2 2 −t1 2 𝑘2 2 x2 2 x1 2 +t1 4 𝑘2 2 x2 2 x1 3/2
= (5-9)
1−t1 2 x1 −t2 2 x1 −𝑘1 2 𝑘2 2 x1 x2 2 +t1 2 t2 2 x1 2
88
Equation (5-9) gives a closed form for the drop port response of the two rings in parallel
based on the application of Mason’s rule.
Considering the case of a single ring in order to validate this approach in the limiting case,
Lp = 0, 𝑘2 = 0
−𝑘1 2 x11/2
G=
1 − t1 2 x1
This is the same as for single ring as presented in [104]. The rest of the model derivation
for an arbitrary number of rings (𝑁 > 2) is presented in the appendix A.
The effect of different parameters on the through and drop port spectral responses of
OADMs are studied as below:
1. An estimation of the separation distance (Lp ) effect is performed. A parallel double
coupled OADM with rings of radii 5 µm (a typical value) is modelled analytically using
(5-9). The drop port response is highly affected by the separation between the resonators.
This comes from the phase accumulation between the dropped signals from the first and
second resonators, which in turn results from the separation distance. The separation
distance should be greater than the ring diameter to ensure that there is no coupling
between rings. The optimum separation [152] is the distance over which the outputs of two
rings are added in phase. Figure 5-4, shows the spectral response of the drop port for two
different separation distances 𝐿𝑝 . In this design the optimum separation is Lp = 0.50462 ×
Lr .
89
Figure 5-4. Analytically calculated spectral response for a 0.05 coupling.
2. For the optimum separation, the effect of changing coupling coefficients in both rings is
simulated analytically and presented in Figure 5-5. Increasing coupling coefficients results
in an increase in the filter bandwidth which can help to drop channels of high data rates
with low level of crosstalk (intra-band crosstalk). However, the OBRR which represents the
level of suppression to adjacent channels (inter-band crosstalk) will be low. The OBRR is
highly affected by the coupling coefficients and for high coupling levels tends to be
unacceptable for WDM filters [107].
90
Figure 5-5. The model sensitivity for different coupling coefficients.
5.3. CST Simulation
A Silicon-On-Insulator ring resonator (Figure 5-1) is modelled using the following
parameters (identical to those used with the SFG approach): ring radius equal to 5 μm,
silicon waveguide with a core refractive index (ncore= 3.47), and silicon dioxide lower
cladding with a refractive index (nclad= 1.47). The upper clad refractive index was equal to
1 (air). The cross section of the silicon waveguide was chosen to ensure a single mode
propagation (width = 460 nm and height = 220 nm) [18] and a 1 𝜇𝑚 thick SiO2 layer is
assumed to ensure minimal substrate leakage losses [8, 9].
The time domain solver results using CST MWS for the parallel coupled ring resonator are
shown in Figure 5-6. In this Figure, S21 and S31 represent through port and drop port
91
spectral responses respectively. The gap separation [g] was 100 nm. The accuracy of the
CST model was first examined against the already modelled filter in [152]. This enables
CST to be used as a reference to validate the analytical model presented in this chapter.
As shown in Figure 5-6, the OBRR is greater than 20 dB, which is similar to that obtained
from the analytical model (Figure 5-5). There is also a comparison between the changes in
the spectral responses with the separation distance in both models; good agreement was
found for both optimum separation and separation equal to half resonator length. This
allows the use of this model to study the effect of different parameters on the overall filter
performance.
92
(a)
(b)
Figure 5-6. CST simulation of a symmetric parallel coupled ring resonator for optimum
separation (a) and half resonator length (b).
93
5.4. Analytical and Simulation Results
In order to mitigate the crosstalk effect in parallel coupled OADMs, both types of crosstalk
(inter and intra-band) should be considered. The proposed model, which has already been
examined against CST MWS, can be used to suggest the optimum design parameters
(separation distance, coupling coefficients, and loss coefficient) that reduce crosstalk and
ensure improved signal integrity.
1. Inter-band crosstalk: Figure 5-5 shows the effect of changing coupling coefficients on
the level of suppression (OBRR). However, that figure shows the change of OBRR for
similar coupling coefficients in both rings. Examining the effect of different coupling
coefficients to obtain optimized values is the aim of this section. The OBRR is calculated
out of resonance to estimate the inter-band crosstalk suppression. Figure 5-7 shows that the
coupling coefficient selection is limited by the OBRR. It is clear that for high coupling
coefficients, the OBRR is less than |20 |dB which is insufficient to suppress the WDM
channels out of resonance.
2. Intra-band crosstalk: Figure 5-8 shows the effect of changing coupling coefficients in
both rings on the level of crosstalk. Crosstalk suppression ratio is highly affected by
coupling coefficients that control the level of drop and through port responses (as discussed
in Chapter three).
94
-18
-19
OBRR [dB] -20
-21
-22 (k )2=0.05
2
-23 =0.06
=0.07
=0.08
-24
=0.09
=0.1
-25
0.05 0.06 0.07 0.08 0.09 0.1
Coupling coefficient of the first ring (k 1)2
Figure 5-7. The OBRR sensitivity to the coupling coefficients.
62
60
Crosstalk at resonance [dB]
58
56
(k2)2= 0.09
54
=0.08
=0.07
52
=0.06
=0.05
50
0.05 0.06 0.07 0.08 0.09 0.1
Figure 5-8. Coupling coefficient effects on the level of crosstalk suppression.
95
A high level of crosstalk suppression is obtained by increasing the coupling coefficients.
However, for efficient design, both crosstalk suppression ratio and OBRR should be within
an acceptable level (greater than |20| dB). Based on Figure 5-7 and Figure 5-8 it can be
seen that, to ensure acceptable levels for inter and intra-band crosstalk, a coupling
coefficient of 0.05 for the second ring can be used with a wide range of coupling
coefficients for the first ring.
For coupling coefficient of 0.05 in the second ring, the effect of losses on the level of
crosstalk suppression was also simulated as shown in Figure 5-9. As expected from the
analytical model, any increase in losses inside the rings will result in a reduction of
crosstalk suppression due to the insertion loss.
Figure 5-9. Crosstalk as a function of losses.
96
Using the analytical model presented in this chapter, estimations of crosstalk bandwidth are
obtained as a function of coupling coefficients, as shown in Figure 5-10. For a
0.05 coupling coefficient for both rings, a bandwidth of about 14 GHz is obtained.
Increasing the value of coupling coefficients leads to an increase in the bandwidth but
decreases the OBRR.
10
x 10
3.5
3
Crosstalk bandwidth [Hz]
2.5
2 (k )2=0.05
2
=0.06
=0.07
1.5
=0.08
=0.09
=0.1
1
0.05 0.06 0.07 0.08 0.09 0.1
Figure 5-10. Crosstalk as a function of coupling coefficients calculated analytically.
To validate the bandwidth obtained using the SFG method, CST MWS was used to
estimate the bandwidth. A time domain simulation for the filter using the optimized
coupling values (calculated using SFG) was implemented and the result is shown in
Figure 5-11.
97
Figure 5-11. Crosstalk bandwidth for an optimal coupling coefficient of 0.05, calculated using CST MWS.
It is clear from Figure 5-11 that the crosstalk suppression bandwidth is 13.16 GHz, which
agrees with that calculated using the SFG model (14 GHz as in Figure 5-10). Increasing the
coupling of both rings to 0.06 will increase the bandwidth of crosstalk suppression to about
21GHz (Figure 5-10) and at the same time, maintain an acceptable level of OBRR
(−23 dB, Figure 5-7). The new coupling coefficients were also modelled using CST, and
the results are shown in Figure 5-12.
The simulation results show that a bandwidth of 20.2 GHz is obtained compared to 21 GHz
from the analytical model. This difference between the calculated bandwidth (which can be
related to meshing in CST) is small enough to allow to rely on the SFG model for crosstalk
bandwidth estimation as a faster and easier approach, compared to the transfer matrix
method [72].
98
Figure 5-12. Crosstalk bandwidth for a 0.06 coupling coefficient calculated using CST MWS.
5.5. Conclusion
In this chapter, the crosstalk performance of a parallel coupled (double) ring resonator
based OADM was studied and modelled. The spectral response was calculated analytically
(using Mason’s rule) and compared with CST MWS generated results for different
separation distances. The SFG model derived in this chapter is sufficiently accurate and
valid to be used for parallel coupled filter analysis. It was used to estimate the coupling
coefficients, in both rings, to ensure an acceptable level of OBRR and increased crosstalk
suppression bandwidth. The small difference between analytically calculated crosstalk
suppression bandwidth (21 GHz) and that obtained numerically using CST MWS (20.2
GHz) provides further validation of the proposed (SFG) calculations. The SFG calculations
99
in parallel coupled OADMs shorten the crosstalk calculation time compared to full-wave
electromagnetic simulation and reduces the computational complexity relative to the
scattering matrix method.
100
CHAPTER SIX
VERTICALLY COUPLED RING
RESONATOR OADM
This chapter proposes a design of small size (single ring) OADM that provides an
increased crosstalk suppression bandwidth to drop 10 Gbps NRZ channels. A vertical
coupled OADM is simulated and the effect of different design parameters on the crosstalk
suppression bandwidth is numerically investigated. A simulation-driven design
optimization procedure is used to determine the design parameters that produce 21 GHz
crosstalk suppression bandwidth.
6.1. Introduction
Ring resonator based add/drop multiplexers (OADMs) and filters are used for adding and
dropping channels entirely in the optical domain. The frequency response of an OADM
depends on the coupling strength (as discussed in Chapter three) [50]. A high coupling
strength will result in a deep notch in the through port response. However, it will reduce
filter selectivity [18]. The coupling strength is highly dependent on the coupling scheme
(lateral and vertical coupling) [103].
101
CHAPTER SIX VERTICALLY COUPLED RING RESONATOR OADMs
In order to design a small size OADM with improved signal integrity, a vertical coupled
“single” ring resonator OADM is simulated and the effect of different design parameters on
the crosstalk suppression bandwidth is numerically investigated. The proposed design
guideline offers a large crosstalk suppression bandwidth with good opportunities for
optimization and control. In a vertically coupled OADM, coupling efficiency between the
evanescent tails of modes (in the bus and bent waveguides) is controlled by the vertical
separation and lateral deviation between waveguides.
The aim of this chapter is to design a small-sized (single ring) vertically coupled OADM
that provides improved crosstalk suppression bandwidth.
1. Vertical coupling in a ring resonator based OADM is introduced and the coupling
coefficient definition in term of ring parameters is presented.
2. Ring parameters effects on the crosstalk suppression ratio are studied and numerically
modelled. Vertical separation and lateral deviation between bus and bent waveguides are
examined to suggest an initial range of parameters for optimization.
3. A simulation-driven design optimization procedure is used to determine the design
parameters that increase the crosstalk suppression bandwidth. The optimization in this
chapter was performed in collaboration with Reykjavik University.
102
This chapter concludes with a design of a vertically coupled OADM that allows for 21
GHz crosstalk suppression bandwidth. More discussion of this design is presented in the
following sections.
6.2. Crosstalk Suppression: Analytical and Simulation
Model
In a vertically coupled OADM, bus waveguides are buried in silicon dioxide material which
will result in a low scattering loss [23]. Moreover, the fabrication process depends on the
well-controlled deposition instead of etching to control the coupling separation [88, 153],
which represents another advantage over the lateral coupling. Coupling coefficients in
vertically coupled structures can be controlled to a fine degree using a number of
parameters such as: vertical separation, lateral deviation and waveguides height. These are
all considered in this chapter.
6.2.1. Analytical Model
The transfer function of the vertically coupled OADM shown in Figure 6-1 is calculated
based on the Coupled Mode Theory (in space domain). A similar approach to that of the
laterally coupled ring resonator used in chapter four is applied [154].
103
The drop port response is given by:
𝑘1 2 𝑘2 2 e−α𝑙 (6-1)
DR =
1 + t1 2 t 2 2 e−2α𝑙 − 2t1 t 2 e−αL cos β𝑙
whereas the through port intensity response can be calculated as:
t1 2 + t 2 2 e−2α𝑙 − 2t1 t 2 e−α𝑙 cos β𝑙 (6-2)

IR =
1 + t1 2 t 2 2 e−2α𝑙 − 2t1 t 2 e−α𝑙 cos β𝑙
Here, 𝑙 is the resonator perimeter, α is the field loss coefficient (the round trip amplitude
reduction exp(−α𝑙)), and β = 2πneff /λ.
Figure 6-1 (a) Vertical coupled ring resonator. (b) Cross section of the bent and bus waveguides.
Based on the coupled mode calculations, the power coupling coefficient 𝑘 2 is related to
structure dimensions as [155-157]:
θo 2
2
(2r + 𝑤𝑟 )2 (6-3)
𝑘 = [sin [( ) K 0 ∫ (cos θ − cosθo )cos 2 θdθ]]
4𝑤𝑟 −θo
104
Where,
2r − 𝑤𝑟 + 2a
θ0 = cos−1
2r + 𝑤𝑟
ωϵ0
K0 = ⋅ (n1 2 − n2 2 ) ∬ f1 ∗ (x, y) ⋅ f2 (x, y) dx. dy
4
f1 ∗ (x, y), and f2 (x, y) are the field profile of bus and bent waveguides, respectively [158].
ω is the angular frequency of light, 𝑤𝑟 is a waveguide width, r is the radius, a is the lateral
deviation and ϵ0 is the free space permittivity.
The crosstalk suppression ratio (XT) is measured as the difference between drop and
through port responses.
XT = 10 ⋅ log DR − 10 ⋅ log IR (6-4)
𝑘 4 e−α𝑙 (6-5)
XT = 10 log
1 + t 4 e−2α𝑙 − 2t 2 e−α𝑙 cos β𝑙
t 2 + t 2 e−2α𝑙 − 2t 2 e−α𝑙 cos β𝑙

− 10log
1 + t 4 e−2α𝑙 − 2t 2 e−α𝑙 cos β𝑙
Since the calculation of crosstalk suppression in (6-5) depends mainly on the coupling
coefficients, which are difficult to calculate from (6-3) without an estimation of the field
profile, a full-wave electromagnetic (EM) simulation is necessary to reliably model the
resonator, and numerically calculate the spectral responses of different ports. The coupling
105
coefficient values are mainly affected by the vertical separation (d), lateral deviation (a),
waveguides height (hg, and hr) and the intermediate layer (n2) refractive index.
6.2.2. CST Simulation
The CST MWS [139] model for an SOI ring resonator is shown in Figure 6-2. Bus
waveguides are modelled using silicon (Si) with refractive index of 3.47, buried in a silicon
dioxide (SiO2) layer with refractive index of 1.47. The ring waveguide is modelled using Si
on the top of SiO2 layer. The cross section of silicon waveguide is (0.34 µm × 0.34 µm).
The ring radius is chosen to be 5 µm (typical value).
Figure 6-2. CST model of a vertically coupled ring resonator OADM.
106
To determine the optimum crosstalk suppression ratio, the time domain solver in CST
MWS is used to perform the numerical calculations for different values of coupling
coefficient (by changing d and a). Lateral deviation is considered as (–a) if the bus
waveguides are moving towards each other. The value of (a) is set initially equal to 0, and
then different values of (d) are simulated. Figure 6-3 shows the through and drop port
responses, and demonstrates the spectral features of the OADM. Insertion loss is calculated
from the drop port response (S31) at resonance, while the OBRR represents the minimum
value of (S31) out of resonance. The crosstalk suppression is calculated as (XT= S31 – S21).
Figure 6-3. The spectral response of CST simulated ring resonator OADM.
107
The results of XT calculations are shown in Figure 6-4, where the value of d that results in
optimum crosstalk suppression and acceptable OBRR is found as 0.3 µm. For 𝑑 = 0.3 µm
and 𝑎 = 0, the crosstalk suppression is found to be 20 dB. However, by keeping (d) and
changing the value of (a), the crosstalk suppression increases to 23 dB with – 25 dB Out-
of-Band Rejection Ratio. The design parameters [a d] for optimal XT are [– 0.001 0.3].
Figure 6-4. Through port, Drop port, and out of band rejection ratio (OBRR) of the ring
resonator as a function of vertical separation for optimized offset (a).
108
6.3. Crosstalk Suppression Bandwidth
Based on the crosstalk suppression bandwidth calculations in [154], the bandwidth over
which a |20| dB XT is maintained can be written as:
c t 2 + t 2 e−2α𝑙 − 0.01k 4 e−α𝑙 (6-6)

∆f = [2πN + cos−1 ]
πneff 𝑙 2t 2 e−α𝑙
Where, N is the mode index of the resonator, and c speed of light in free space.
It is clear from (6-6) that the crosstalk suppression bandwidth depends on coupling
coefficients and the effective refractive index. Analytically, Figure 6-5 shows the effect of
changing power coupling coefficient on the useful bandwidth. Based on Figure 6-5, it is
clear that to drop a 10 Gbps NRZ signal (20 GHz crosstalk suppression bandwidth is
required) the required coupling coefficient is 0.625. For a laterally coupled single ring
resonator this value refers to a strong coupling which means a narrow gap between the bus
and bent waveguides (as discussed in Chapter four).
109
Figure 6-5. Analytically calculated crosstalk suppression bandwidth as a function of coupling coefficient.
6.4. Optimization Method
In this chapter, the crosstalk suppression bandwidth is maximized using an EM-simulation-
driven design optimization [141]. Two design scenarios, with (i) two, and (ii) four design
parameters, are considered. The design variable vectors for these two scenarios are as
follows:
 x = [a d]T
 x = [a d hg hr]T
110
For the first scenario, the parameters hg and hr are fixed to 0.34 m. Design variable ranges
are as follows: 0.0025 m  a  0.0025 m, 0.1 m  d  0.35 m, 0.1 m  hg  0.4
m, 0.1 m  hr  0.4 m.
The optimization process design specifications are concerning S21 and S31 as follows:
 Maximize the bandwidth (here, denoted as B); the minimum required B is 20 GHz;
 Ensure high OBRR, specifically, |S31|  20 dB.
The above design problem is formulated as a nonlinear minimization task of the form
x*  arg min H ( R( x)) (6-7)

x
Where R(x) is a response vector of the EM-simulated ring model, H is an objective
function that encodes the specification requirements, whereas 𝒙* is an optimal design to be
found. The first of the aforementioned design goals is treated as the primary objective. The
second goal is handled using a suitably defined penalty function. Thus, the objective
function takes the form:
𝑚𝑎𝑥{𝑚𝑖𝑛|𝑆31 | + 20,0} 2 (6-8)

H(𝐑(𝑥)) = −B(𝑥) + 𝛽 ⋅ [ ]
20
Where  is a penalty factor (here,  = 10). Formulation (6-8) maximizes the bandwidth
while penalizing the designs for which the second goal is not satisfied (i.e., |S31| > 20 dB
at its minimum).
111
To solve (6-7), a pattern search algorithm [159] is used. Pattern search is a derivative-free
stencil-based optimization technique where the search is constrained to a grid (here,
rectangular), which is iteratively refined as necessary (i.e., when the search on a current
grid fails to improve the design). A specific version of the method is utilized here, a grid-
constrained line search and a few other modifications [160] to reduce the computational
cost of the optimization process. The use of pattern search technique is motivated by the
fact that the EM-simulation model is rather noisy. For this kind of problems the use of
gradient-based methods is generally not recommended [141].
The initial design is xinit = [0 0.275 0.34 0.34]T. The bandwidth for this design is around 9
GHz. Upon optimization, it turns out that the constraint |S31| ≤ 20 dB is too strict, and 20
GHz bandwidth cannot be achieved for either design scenario (two- and four-variable case).
For the two-variable case, the maximum obtained bandwidth (while keeping |S31| ≤ 20 dB)
is 11.9 GHz, and the optimal parameters are [– 0.002 0.2577]T. For the four-variable case,
the obtained bandwidth is wider (13.7 GHz) but still below the requested threshold of 20
GHz.
As the intra-band crosstalk results in a higher system performance degradation compared to
inter-band crosstalk (which is easier to remove in the optical receiver) [67, 73], the
intention is to increase the bandwidth of the intra-band crosstalk suppression. Another,
somehow relaxed OBRR constraint (|S31|≤ – 15dB) is applied in the optimization; the
obtained results show an improvement in the bandwidth calculations. For two-variable
optimization, the obtained bandwidth is 19.8 GHz (see Figure 6-6) with the corresponding
112
parameter setup xopt.1 = [0.0025 0.2145 0.34 0.34]T. For four-variable case, the
optimization process yields the design xopt.2 = [0.0023 0.2152 0.34 0.345]T and the
corresponding bandwidth is 21 GHz (see Figure 6-7). The design cost for the two- and
four-variable case is 36 and 80 EM analyses of the structure, respectively. The
optimization results are summarized in Table 6-1.
Table 6-1. Optimization results
Lateral Vertical Bus WG Bent WG OBRR XT

deviation separation (d) height (hg) height (hr) Constraint bandwidth
(a) [µm] [µm] [µm] [µm] [dB] [GHz]
– 0.002 0.2577 0.34 0.34 ≤ – 20 11.9
– 0.0015 0.2471 0.3406 0.3456 ≤ – 20 13.7
– 0.0025 0.2145 0.34 0.34 ≤ – 15 19.8
– 0.0023 0.2152 0.34 0.345 ≤ – 15 21.0
The design setup Xopt.2 presented in Table 6.1 results in a 21 GHz crosstalk suppression
bandwidth. However, the dimensions of a and d seem to be difficult to realise in the
practical implementation. Therefore, examining of different sets of a and d is performed in
order to reduce the fabrication sensitivity resulted from using very small dimensions. A
new parameter setup is obtained where the dimensions are more realistic and at the same
time result in a 20GHz (which is sufficient to drop 10 Gbps modulated channel). The new
parameter setup is X= [−0.002 0.21 034 0.345 ] which results in a 20.1 GHz XT
bandwidth.
113
-5
|S21|, |S31| [dB]
-10
-15
-20
|S21|
-25
|S31|
-30
190.5 191 191.5 192 192.5 193 193.5
Frequency [THz]
(a)
-5
|S21|, |S31| [dB]
-10
-15
-20
|S21|
-25
|S31|
-30
190.9 190.95 191 191.05 191.1 191.15 191.2 191.25 191.3 191.35 191.4
Frequency [THz]
(b)
Figure 6-6. Initial (thin lines) and optimized (thick lines) responses for design case (i) (two
design variables): (a) |S21| and |S31| for 190.5 to 193.5 THz range, (b) magnification
around 191 THz. Optimized 20dB-bandwidth is 19.8 GHz.
114
-5
|S21|, |S31| [dB]
-10
-15
-20
|S21|
-25
|S31|
-30
190.5 191 191.5 192 192.5 193 193.5 194 194.5
Frequency [THz]
(a)
-5
|S21|, |S31| [dB]
-10
-15
-20
|S21|
-25
|S31|
-30
192 192.05 192.1 192.15 192.2 192.25 192.3
Frequency [THz]
(b)
Figure 6-7. Initial (thin lines) and optimized (thick lines) responses for design case (ii) (four
design variables): (a) |S21| and |S31| for 190.5 to 193.5 THz range, (b) magnification
around 192THz. Optimized 20dB-bandwidth is 21 GHz.
115
6.5. Conclusion
The crosstalk suppression in a vertically coupled ring resonator OADM was investigated
and numerically simulated. This chapter was started by proposing design parameters that
increase the crosstalk suppression ratio. Vertical separation and lateral deviation between
bus and bent waveguides were used to maximize the through port notch of the filter. Then,
a pattern search optimization algorithm was used to maximize the crosstalk suppression
bandwidth in a single ring based OADM. This approach allows using an electromagnetic
simulation to perform the optimization and provides the coupling region dimensions
(vertical separation and lateral deviation). Design parameters that produce a 21 GHz
crosstalk suppression bandwidth were proposed.
116
CHAPTER SEVEN
GRATING-ASSISTED RING RESONATOR
OADM
In this chapter, resonance splitting induced by sidewall roughness in silicon waveguides is
exploited to increase the crosstalk suppression bandwidth over the value that would be
expected from a smooth-walled resonator. A general form for the spectral response of a
rough-walled ring resonator is derived analytically using the space and time domain
Coupled Mode Theory. Verification against results generated from numerical modelling of
a randomized rough-walled ring resonator is performed. A design which is more
controllable during manufacture than a purely randomized variation is proposed. This
chapter concludes with the design of a grating-assisted single ring resonator that provides
28 GHz crosstalk suppression bandwidth.
7.1. Introduction
Optical add/drop multiplexer analyses exhibit a high dependency of the spectral response
upon the characteristics of coupling regions (gap separation and the length of coupling
regions) [50]. Theoretical analysis shows a symmetric spectral response at different
resonant wavelengths. However, experimental results of ring resonators reveal a high
difference in the through and drop port responses at consecutive resonances [161].
117
CHAPTER SEVEN GRATING-ASSISTED RING RESONATOR OADMs
The frequency response discrepancy results from the sidewall roughness induced back
reflection inside the ring [116]. Sidewall roughness leads to a splitting of response at
resonance due to the mutual coupling between the forward and backward propagated modes
[18].
Estimation of mutual coupling and reflection coefficients depends mainly on the nature of
the sidewall roughness. In most cases sidewall roughness was considered as a random
perturbation. Therefore, the resulting back reflection is treated as a stochastic average
[114]. However, it was shown in [71] that sidewall roughness can be treated similarly to a
structure comprising gratings with rectangular shapes. The corrugation can consist of a
group of ridges with a similar length and period. A semi-periodic arrangement of 30 −
50 𝑛𝑚 ridges was formed using the electron beam lithography, and the backscattering level
was calculated as a deterministic function of wavelength [71].
The possibility of predefining backscattering levels by controlling the shape of quasi-
grating in the fabrication process allows for new applications of grating-assisted ring
resonators [71, 161, 162]. Using Fibre Bragg Grating calculations [70], back reflection
effect can be controlled by changing gratings dimensions (grating length, period, and
number of ridges).The dual-mode filter model that is used in microwave engineering field
[163] has been exploited in the field of optical photonics to improve filter performance by
exploiting the controllable reflectivity [118, 164].
118
This chapter aims to:
1. Propose a general model for rough-walled ring resonators.
2. Use the controllable reflectivity resulting from a periodic variation of the sidewall
roughness to increase the crosstalk suppression bandwidth.
1. A mathematical model based on the time domain Coupled Mode Theory (CMT) is
presented. This model allows for a complete characterization of all parameters of the
ring including back reflection.
2. An equivalent structure of the rough-walled ring is proposed and the space domain CMT
calculations are used to model different ports spectral responses.
3. The accuracy of the time and space domain CMT calculations is examined against
existing experimental results. A rough-walled ring resonator is modelled using CST
MWS to simulate the spectral response of different ports.
4. Controllable reflectivity resulting from semi-periodic gratings is modelled and validated
using the ASPIC design simulator [165] to examine its accuracy. A general model of a
grating-assisted ring resonator is derived.
5. An optimization approach based on the goal maximization algorithm (in Excel) is
performed to calculate all parameters that maximize the crosstalk suppression
bandwidth.
This chapter proposes a general solution for rough-walled ring resonators modelling, as
well as a particular solution to maximize the crosstalk suppression bandwidth. It concludes
119
with a design that provides a 28 GHz crosstalk bandwidth. This bandwidth can be used to
drop 10 Gbps NRZ channels with a low level of crosstalk.
7.2. Coupled Mode Analysis
In this section, the analysis of mutual coupling between the forward mode (deliberately
excited by the bus waveguide) and back reflected mode (induced by sidewall roughness) is
presented and analytically modelled using CMT.
7.2.1. Time Domain Analysis
Referring to Figure 7-1, if the incident wave at the input port is Si and considering that
there is no added signal, the amplitude of forward mode inside the resonator is 𝑎(𝑡) and the
sidewall roughness induced backward mode is 𝑏(𝑡). The mutual coupling between 𝑎(𝑡)
and 𝑏(𝑡) depends mainly on the reflection coefficient R.
120
Figure 7-1. Forward and backward modes in a rough-walled ring resonator add/drop filter.
Starting with the time domain CMT analysis presented in [93, 116], the change rate
equation of the energy stored in the ring (forward mode) is modified to include 𝑏(𝑡) as
below:
da(t) 1 (7-1)
= (jω0 − ) a(t) − j𝑘1 Si − jub(t)
dt τ
𝑉𝑔
where, u = √R ⋅ is the mutual coupling, 𝑉𝑔 is the group velocity, 𝑙 is the perimeter of the
𝑙
1
resonator, and is the decay rate of energy inside the resonator (determined by coupling
τ
coefficient and losses in the ring).
Similarly, the change rate equation of back reflection mode energy is modified as:
db(t) 1
= (jω0 − ) b(t) − jua(t)
dt τ
(7-2)
121
The power transfer characteristics are calculated at a steady state by considering the input
signal with time dependency ejωt . From (7-1) and (7-2), 𝑎(𝑡) and 𝑏(𝑡) are:
−j𝑘1 Si − jub(t) (7-3)

a(t) =
A
1 (7-4)
A = j(ω − ωo ) +
τ
−jua(t) (7-5)
b(t) =
A
And from (7-3) and (7-5), (7-6) is obtained:
−j𝑘1 A (7-6)
a(t) = ⋅S
A2 + u2 i
Assuming the propagation constant in the bus waveguide of length l is β, different port
responses can be expressed as below:
i. Through port spectral response
St = ejβl (Si − j𝑘1 a(t)) (7-7)
2
St 2 𝑘1 2 A (7-8)
| | = |1 − 2 |
Si A + u2
ii. Drop port spectral response
Sd = −j𝑘2 a(t) (7-9)
122
Sd 2 𝑘1 𝑘2 A 2 (7-10)
| | =| 2 |
Si A + u2
iii. Back reflection at the add port
Sa = −j𝑘2 b(t) (7-11)
−𝑘1 𝑘2 u (7-12)
Sa = a(t)
A
Sa 2 𝑘1 𝑘2 u 2 (7-13)
| | =| 2 |
Si A + u2
At resonance, the three ports transmission spectra are as below:
2
1 𝑘1 2
2
(7-14)
2 [u + 2 − τ ]
St τ
| | =
Si 1 2
[u2 + ]
τ2
2
𝑘1 2 𝑘2 2 (7-15)
Sd τ2
| | =
Si 1 2
[u2 + 2 ]
τ
And
Sa 2 𝑘1 2 𝑘2 2 u2 (7-16)
| | =
Si 1 2
[u2 + 2 ]
τ
Different port responses at resonance can be written as below:
S 2 S 2 S 2
Tho = |St| , Dro = | Sd| , and Reo = | Sa |
i i i
123
Where, Tho , Dro and Reo are the through, drop and back reflection levels at resonance,
respectively.
From (7-15) and (7-16):
Dro 1 (7-17)
= 2 2
Reo τ u
Then
1 Reo (7-18)
u2 = ⋅
τ2 Dro
To calculate τ, the ratio of drop (7-10) and back reflection (7-13) responses is taken at the
1
frequency of Re = 2 Reo , which is denoted as f1 . After some rearrangements:
(7-19)
1 2Dr
τ= √ −1
∆ω Dro
Dr is the value of the drop response at f1 and ∆ω is the frequency difference between f1 and
the resonance frequency.
By calculating τ, the value of reflection coefficient is easily calculated from (7-18) as:
(∆ω)2 𝑙 2 Dro Reo (‎

7-20)
𝑅= 2
⋅
𝑣𝑔 √2Dr − Dro
while the values of coupling coefficients can be calculated from (7-14) and (7-15) as below:
𝑙 (Reo + Dro ) (7-21)

𝑘1 2 = ⋅ ⋅ [1 − √𝑇ℎ𝑜 ]
𝑣𝑔 τ 𝐷𝑒𝑜
124
𝑙 (Reo + Dro ) (7-22)

𝑘2 2 = ⋅
𝑣𝑔 τ [1 − √𝑇ℎ𝑜 ]
And the power loss coefficient 𝑘𝑝 2 can be calculated based on the calculation of [166] as:
2𝑙 (7-23)
𝑘𝑝 2 = [ ] − 𝑘1 2 − 𝑘2 2
𝑣𝑔 τ
Then the loss factor 𝛼 can be calculated as:
1 (7-24)
𝛼= [−10 log(1 − 𝑘𝑝 2 )]
𝑙
7.2.2. Space Domain Analysis
Although the reflection is distributed along the ring, it can be considered as a lumped
scattering point without loss of generality [113]. The lumped scattering point is defined by
the reflection coefficient (𝐾𝑟 2 ) and transmission coefficient (𝑡𝑟 2 ). Figure 7-2 (a) shows the
rough-walled single ring model. An equivalent structure is proposed, shown in Figure 7-2
(b), in which the reflection is considered to be coming from a virtual mirror image of the
ring. This illustrates the generation of the counter-directional mode inside the ring as a
result of sidewall roughness.
125
Figure 7-2. a. The schematic diagram of a rough-walled ring resonator, b. Its equivalent structure.
The drop port (Sd), through port (St) and back-reflection (Sback) responses of the single ring
resonator add/drop filter are calculated by writing loop equations at different nodes inside
the ring as follows:
𝑆𝑡 = −𝑗𝑘1 𝑎6 + 𝑡1 𝑆𝑖
Sd = −j𝑘2 a4 (7-25)
Sback = −j𝑘2 t 2 𝑏2 x 3/2
𝑎6 = 𝑎5 𝑥1/2 (7-26)
where, 𝑥 = 𝑒 −𝛼𝑙−𝑗𝛽𝑙 and,
𝑎5 = 𝑡2 𝑎4 = 𝑡2 𝑎3 𝑥1/2 𝑒 −𝑗𝜃 (7-27)
with 𝜃 the phase position of the lumped scattering point.
126
𝑎3 = −𝑗𝑘𝑟 𝑏3 + 𝑡𝑟 𝑎2 (7-28)
𝑏3 = 𝑏2 𝑡1 𝑡2 𝑥 (7-29)
𝑏2 = −𝑗𝑘𝑟 𝑎2 + 𝑡𝑟 𝑏3 (7-30)
𝑎2 = 𝑎1 𝑒 −𝑗𝜃 = (𝑡1 𝑎6 − 𝑗𝑘1 𝑆𝑖 )𝑒 −𝑗𝜃 (7-31)
The drop port response in the presence of sidewall roughness is derived using the above
equations as:
−𝑘1 𝑘2 [𝑡𝑟 − 𝑡1 𝑡2 𝑒 −𝑗∅ ]𝑒 −𝑗∅/2

𝑆𝑑 =
1 − 2𝑡1 𝑡2 𝑡𝑟 𝑒 −𝑗∅ + 𝑡1 2 𝑡2 2 𝑒 −2𝑗∅ (7-32)
Where, ∅ = 𝛼𝑙 + 𝑗𝛽𝑙 is the propagation constant around the ring, 𝛼 and 𝛽 are the loss and
phase coefficients respectively. 𝑘1 and 𝑘2 are the coupling coefficient between bus and ring
waveguides, 𝑙 is the perimeter of the ring and 𝑡𝑟 = √1 − 𝑘𝑟 2 .
The through port response is calculated as:
𝑡1 − 𝑡𝑟 𝑡1 2 𝑡2 𝑒 −𝑗∅ − 𝑡𝑟 𝑡2 𝑒 −𝑗∅ + 𝑡1 𝑡2 2 𝑒 −2𝑗∅

𝑆𝑡 = (7-33)
1 − 2𝑡1 𝑡2 𝑡𝑟 𝑒 −𝑗∅ + 𝑡1 2 𝑡2 2 𝑒 −2𝑗∅
And finally, the reflected signal at the add port as a result of the backward propagated mode
induced by the sidewall roughness is given by:
127
𝑗𝑘𝑟 𝑘1 𝑘2 𝑡1 𝑒 −𝑗3∅/2
𝑆𝑏𝑎𝑐𝑘 =
1 − 2𝑡1 𝑡2 𝑡𝑟 𝑒 −𝑗∅ + 𝑡1 2 𝑡2 2 𝑒 −2𝑗∅ (7-34)
The above equations provide the spectral features of a rough-walled ring resonator and can
be used to obtain the spectral responses of different ports based on the parameters
calculated from the time domain model. This allows for a general modelling of a rough-
walled ring resonator and reproducing the experimental results without the need for curve
fitting.
These models (time and space domain) are validated first against the experimental results
presented in [111].
1. The analytical model (time domain) represented by equations (7-20) to (7-23) is used to
extract the coupling coefficients from the experimentally calculated spectral response
7-3 (b). The calculated parameters are: 𝑘1 2 =

presented in [111] and shown in Figure ‎
4.8%, 𝑘2 2 = 1.76%, 𝑡𝑟 = 0.9991 and the round trip loss = 0.9639.
2. The space domain model is used to plot the spectral response and reproduce the
experimental results using coupling coefficients (𝑘1 2 , 𝑘2 2 , 𝑡𝑟 , and loss coefficient)
calculated in step 1, as shown in Figure 7-3 a.
A comparison between Figure ‎

7-3 (a) and (b) shows a good agreement between the
proposed models and experimental results, and allows for the use of these models for filter
performance optimization in terms of crosstalk and signal integrity. For further validation, a
rough-walled ring resonator is modelled using CST MWS and the proposed models are
128
used to extract different resonator parameters from the simulation result, as will be shown
in the next section.
0
1557.8 nm
k2=0.0418
-5 k2=0.0176
tr=0.9991
INTENSITY[dB]
round trip
-10 loss=0.9639
-15
Through port responsee

-20
Drop port response
Back reflection
-25
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
Wavelength difference from resonance [nm]
A b
Figure 7-3. a. Ring resonator response analytically modelled using time and space models. b. Experimental
(line) and analytical (dot) results presented in [111].
7.3. CST Validation
CST MWS [139] is used to model a ring resonator with random sidewall roughness. The
rough-walled ring is first created as a solid model programmatically using Ruby code [167].
The ring was assembled from cuboids. There were two types of cuboid: those that were
narrow, which just had side and top/bottom faces, and those which were wide, which also
had partial front and back faces where they joined the narrow cuboids. The narrow cuboids
were assembled into the bulk of the ring, and the wide cuboids were used to create the
ridged parts of the ring. The cuboids subtended about 1 degree at the centre of the ring for
the smooth parts. These cuboids were collections of 8 points, 12 in the case of the wide
129
cuboids, with an associated list declaring how they were to be wired into faces so that the
normals would face outwards, i.e. the points were listed clockwise around the normal
vector. This was done with triangular meshing for portability. These were joined together to
create one .obj file using Ruby code, and then imported into CST as an object file. The
Ruby code is presented in appendix B. The CST MWS model of the ring resonator based
add/drop filter is shown in Figure 7-4.
Figure 7-4. CST model of sidewall roughness in a single ring resonator add /drop filter.
In the electromagnetic model, the refractive index of the silicon waveguide is 3.47, and that
of the 1 µm silicon dioxide substrate is 1.47 [146]. The upper cladding is air. The ring
radius is 8µm. The cross-section dimensions of the input-output silicon waveguides are
(0.5 µm width × 0.22𝜇𝑚 height) to ensure a single mode propagation in the bus
waveguides [18]. Coupling coefficients are determined by the separation between the ring
and bus waveguides, which are taken as 60 nm and 160 nm for the input and output
waveguides, respectively. These values are chosen to ensure the resonance splitting.
130
Figure 7-5 shows the spectral responses for different ports of a rough-walled ring resonator.
S21, S31, and S41 represent the through, drop and back-reflection responses, respectively.
Figure 7-5. CST frequency response of a rough-walled Ring Resonator.
The analytical model (time domain) is used, first, to extract the modelled rough-walled ring
parameters (coupling, reflection, and loss coefficients) from the simulation result. By using
equations (7-20) to (7-23), the ring parameters are calculated as below:
𝑘1 2 = 10.774%
𝑘2 2 = 1.422%
𝑡𝑟 = 0.998
And
𝑒 −𝛼𝑙 = 0.986
The second step for validation is to put the above obtained values in (7-32) and (7-34) to
131
obtain the spectral responses. Analytically calculated spectral responses of the through,
drop and back-reflection ports are plotted in Figure 7-6 using Matlab code in combination
with CST simulation results to show the validity of the time and space domain calculations.
This provides an extra validation for the proposed analytical models, and allows for using it
to examine the effect of back reflection on the crosstalk suppression and crosstalk
suppression bandwidth.
Figure 7-6. CST (solid) and analytically (dotted) modelled spectral response for a rough-walled ring
resonator.
The effect of back reflection on the crosstalk suppression can be estimated by changing the
reflection coefficient (tr) and calculating the difference between drop and through port
responses at resonance. Figure 7-7 shows clearly that, increasing back reflection coefficient
(reduction of 𝑡𝑟 ) will result in a strong splitting of the response. In a single ring resonator, a
double minimum and single maximum will appear in the through port response as a result
of back reflection as shown in Figure 7-7, and at the same time a double maximum and
132
single minimum will appear in the drop port response. Crosstalk suppression is the
difference between the single minimum of the drop port and the single maximum of the
through port responses. Keeping the crosstalk suppression higher than the required level of
adequate filter performance (≥ |20| dB [23]), for as wide a bandwidth as possible, means
improved crosstalk suppression bandwidth. This can be achieved by controlling the
reflectivity of sidewall roughness in order to propose a filter design which allows for
dropping channels with a low level of crosstalk.
Figure 7-7. The effect of back-reflection coefficient on the through and drop port response.
133
7.4. Controllable Reflectivity
Based on Bragg grating reflectivity calculations [70], a perturbation of the refractive index
due to the variation of waveguide width will result in a generation of backward propagated
mode inside the waveguide. The mutual coupling between forward and backward modes
will lead to the occurrence of resonance splitting due to power transfer between modes.
Figure ‎
7-8 shows single and double gratings.
Figure 7-8. (a) Single and (b) double gratings.
The power reflection coefficient R of a single grating can be expressed as [168]:
𝐾 2 𝑠𝑖𝑛ℎ2 (𝑆𝐿)
𝑅 = |𝑟0 |2 = (‎7-35)
𝛿 2 𝑠𝑖𝑛ℎ2 (𝑆𝐿) + 𝑆 2 𝑐𝑜𝑠ℎ2 (𝑆𝐿)
134
where r0 is the field reflection coefficient and K is the coupling coefficient of the forward
and backward modes which is expressed as:
π∆neff
K= (7-36)
λ
2πneff π
δ is the detuned propagation constant (δ = − ʌ ), ʌ is the grating period, L is the
λ
grating length and S = √K 2 − δ2 . Based on (7-35), the reflectivity is dependent on the
change of the effective refractive index, grating length, and grating period. The calculations
in this section aim to increase the reflectivity by examining different parameters. The
effective refractive index of the SOI waveguide is 2.55 and a uniform change of the
effective refractive index over the grating is considered with ∆neff = 0.5 [18].
The effects of different parameter are examined as follows:
1. The grating length effect: Figure 7-9 shows that increasing the length of grating will
result only in increasing the changing rate of the reflectivity over the wavelength range
(around 1550 𝑛𝑚). In this case the range of wavelengths is 1540-1560 𝑛𝑚 and the best
grating length (as shown in Figure 7-9) is 6500 𝑛𝑚 since it gives relatively high reflectivity
over the wavelength range. A grating period of 100 𝑛𝑚 and duty cycle of 50% (duty cycle
is the ratio of ridge width to the grating period) are used.
135
Figure 7-9. Grating length effect on the reflectivity for L1=6500 nm, L2= 13000 nm and L3= 19500 nm.
2. The grating period effect: for a grating length of 6.5 μm and the same duty cycle,
Figure 7-10 shows an increase in the reflectivity with an increasing grating period. Based
on the diffraction theory, the Bragg wavelength is (𝜆𝐵𝑟𝑎𝑔𝑔 = 2 ⋅ 𝑛𝑒𝑓𝑓 ⋅ ʌ) [71]. Therefore,
increasing the grating period will increase 𝜆𝐵𝑟𝑎𝑔𝑔 and makes it close to the wavelength
range of interest.
136
Figure 7-10. The grating period effect on the reflectivity.
3. Number of reflectors effect: to increase the level of reflectivity two or more gratings can
be used separated by a distance Lr which should be chosen to ensure a proper phase change
between reflected modes from each reflector. If double gratings are used, as shown in
Figure 7-8 b, the overall reflectivity will be a combination of the contributions of each
reflector. However, when adding the two reflectivities, r0 and r1 , a closed loop will be
formed between the two reflectors. Using the Signal Flow Graph method [149] the overall
reflectivity of two gratings:
r0 + r1 e−j2βLr
r0 + r1 = (7-37)
1 + r0 r1 e−jβLr
137
However, if the number of gratings is increased to be three, for example, the total
reflectivity will be more due to the number of reflectors. The total reflectivity of three
gratings is:
r0 + r1 e−j2βLr1 + r2 e−l2β(Lr1 +Lr2 ) + r0 r1 r2 e−j2βLr2

r0 + r1 + r2 =
1 + r0 r1 e−j2βLr1 + r1 r2 e−j2βLr2 + r0 r2 e−l2β(Lr1 +Lr2)
(7-38)
where Lr1 and Lr2 are the separations between gratings. Figure 7-11 shows the overall
reflectivity of a single, double and three gratings.
Figure 7-11. Changing the reflectivity with increasing the number of gratings.
These results are validated first using ASPIC design software [165]. ASPIC is a frequency
domain simulator, it calculates the results by assembling the scattering matrix of each
component in a single large matrix (based on the circuit topology) then uses it to find the
optical field in each node of the OADM [169, 170]. ASPIC is a model based simulation
138
software and approaches simulation differently to the physically based CST MWS
simulation software. Having verified its performance, the validation allows the use the
model for OADM performance optimization. Figure 7-12, shows the effect of using single,
double and three grating. It is shown in Figure 7-12 (b) that, increasing the number of
gratings will result in an increase of back reflection as in [71].
a b
Figure 7-12. ASPIC model for three gratings (a), and the reflectivity as a function of wavelength (b) for
single grating (blue), double gratings (green) and three gratings (red).
4. The effect of changing the separation between gratings can be seen in Figure 7-13. To
ensure high reflectivity, the space between the gratings should correspond to a 𝜋 phase
shift. The total reflectivity is hence strongly related by the length of the waveguides
between the gratings. Therefore, when the gratings are added to the ring, the ring radius
should be optimized in order to ensure a proper separation to maximize the reflection.
139
Figure 7-13. The effect of separation between gratings for three gratings.
7.5. Grating-Assisted Single Ring
In this section, a design of grating-assisted ring resonator OADM that provides wider
crosstalk suppression bandwidth is presented as follows:
Step 1: Based on the calculations of crosstalk suppression bandwidth [154], the coupling
and back reflection coefficients that that maximize the crosstalk suppression bandwidth can
be calculated using (7-32) and (7-33). The optimization process starts by calculating the
difference between drop and through port responses (crosstalk suppression), over the range
of frequencies of one resonance. Different sets of coupling and reflection coefficients are
used. For each set of the coefficients, the bandwidth over which the crosstalk suppression
ratio (S31−S21) exceeds |20| dB threshold is calculated. The goal maximization algorithm
140
in Excel is used to optimize the values of the coupling and reflection coefficients that
produce a maximum crosstalk suppression bandwidth.
Step 2: In addition to the coupling coefficient optimization, the ring radius needed to be
selected to match the resonance wavelength with the required value of reflectivity. The
separation between the three gratings (as discussed in 7.4 step 4) is calculated as (𝐿𝑟 =
((𝑙 − 3 × 𝐿))/3), where 𝑙 is the ring perimeter. To maximize the reflectivity, the separation
between the gratings should be optimized (through the proper choice of ring radius). A
general model that combines all the parameters (coupling coefficients, grating length,
number of gratings, grating period and ring radius) is used. The optimization approach is
performed for two values of grating period (100 and 120 mm) since these two values
provide increased reflectivity as shown in Figure 7-10.
For an asymmetric coupled ring resonator, the optimized ring parameters for crosstalk
bandwidth maximization are obtained as follows: The power coupling coefficients 𝑘1 2 =
0.2258, 𝑘2 2 = 0.0329, and the reflection coefficient 𝑡𝑟 = 0.9914. Figure 7-14 shows the
spectral response for different ports using the optimized parameters. The maximum
crosstalk bandwidth obtained is 28 GHz, which is sufficient to drop a channel of 10 Gbps
NRZ transmission with low level of crosstalk. Based on step 2 above, the back-reflection
coefficient obtained above is used with the grating model [71] to produce the length of the
quasi-gratings inside the ring.
The dimensions for the grating-assisted ring resonator that produces 28 GHz crosstalk
suppression bandwidth are found analytically as follows:
141
1. Three gratings along the ring of 9.64 𝜇𝑚 to be used, the length of each grating is
6.5𝜇𝑚.
2. The period of ridges in each grating is 0.1𝜇𝑚.
Based on the above results, a single ring resonator of 9.64𝜇𝑚 radius, comprising of three
gratings of total length 19.5 𝜇𝑚, each grating has 65 ridges, is shown analytically to
provide a crosstalk bandwidth of 28 GHz. To achieve a similar crosstalk suppression
bandwidth using a smooth-walled ring resonator, the coupling coefficient would need to
exceed 0.625 [154], which would reduce the selectivity of the filter and increase the
crosstalk resulting from the adjacent channels. A similar bandwidth requires the use of a
double ring resonator [154], which means increasing the filter size and reducing the
integration density. To this extent it is shown analytically that a grating-assisted single ring
resonator can be used to drop higher data-rate signals compared to a smooth-walled
resonator. The ASPIC simulator results for a single ring resonator with three gratings are
shown in Figure 7-15, which shows a good agreement with that of [164] where the
resonance splitting is clear and |20| dB crosstalk suppression is maintained for a wide
crosstalk suppression bandwidth.
142
a b
Figure 7-14. a. The spectral response of a single ring resonator (using optimized parameters that maximize
crosstalk bandwidth), b. Schematic of a grating-assisted OADM.
Figure 7-15. The three port response for a grating assisted ring resonator (ASPIC simulated results).
143
7.6. Conclusion
Resonance splitting induced by sidewall roughness in a single ring resonator was studied
and exploited to increase the crosstalk suppression bandwidth. An equivalent structure of
the rough-walled ring resonator was proposed. The spectral responses of different ports
were defined mathematically and validated against experimental results. These models
provide a simple and direct approach to calculate all ring parameters based on the simulated
spectral response without the need for curve fitting. A method of optimizing the number of
grating groups and the length of individual gratings for a given required performance is
proposed. A design of grating-assisted ring resonator was proposed based on semi-periodic
sidewall roughness. This chapter showed that a single ring with three gratings, each grating
measuring 6.5 𝜇𝑚 in length, is capable of providing 28 GHz crosstalk suppression
bandwidth.
144
CHAPTER EIGHT
CONCLUSIONS AND FUTURE WORK
In this chapter, a summary of the thesis is presented, results are discussed and
recommendations for further work are proposed. The conclusions are provided in
Section8.1, while the suggestions for future works are listed in Section 8.2.
8.1. Conclusions
In this thesis, the topic of “Optical” EMC was approached by studying the crosstalk in all-
optical networks; in the particular case of optical ring resonator based add/drop
multiplexers (OADMs) and filters. Optical EMC was of relevance because OADMs are
used in all-optical networks to introduce and drop channels in WDM nodes. Crosstalk in
ring resonator based OADMs results from the adjacent channels (inter-band crosstalk) and
the residual of new added channel (intra-band crosstalk). Crosstalk was mitigated either by
increasing the number of rings (to improving filter response) or increasing the through port
notch depth (increasing the crosstalk suppression ratio). Increasing the crosstalk
suppression ratio leads to high crosstalk suppression in a narrow frequency band around the
resonance frequency, while increasing the number of rings results in an increase in the filter
size conflicting with the goal of greater device density. The integrated solutions to meet
145
CHAPTER Eight CONCLUSIONS AND FUTURE WORK
WDM network requirements require the use of small-size filers that provide
adding/dropping of the modulated channels with low level of crosstalk and improved signal
integrity.
In this thesis,
 Defining “optical” EMC in small-sized OADMs for WDM networks was the main
motivation for this research.
 The key research question in this thesis was: how to improve signal integrity and
mitigate the crosstalk effect in a small-sized OADMs in order to enhance the optical
EMC in all-optical networks and contribute to the increase in integration scalability?
It was answered by:
1. Using SOI ring resonator based OADMs in order to contribute to the increase in
integration scalability (compared to the AWG and FBG based OADMs).
2. Increasing the crosstalk suppression bandwidth rather than the crosstalk suppression
ratio. Crosstalk suppression bandwidth was defined as the bandwidth over which
the level of crosstalk suppression ≥ |20| dB. The bandwidth of modulated channels
is mainly affected by the data rate and the transmission technique. For example, 10
Gbps of NRZ transmission requires 20 GHz bandwidth. Therefore, the crosstalk
suppression bandwidth needs to be more than 20 GHz to ensure adding/ dropping of
this channel with a reduced level of crosstalk.
146
 The objectives were:
1. To exploit the resonance splitting that occurs due to inter-ring coupling
coefficients in a series coupled OADM.
2. To enhance the crosstalk suppression bandwidth by optimizing ring parameters
in a parallel coupled OADM.
3. To optimize coupling coefficients in a vertically coupled ring resonator OADM
to increase the crosstalk suppression bandwidth in a small-sized OADM.
4. To exploit the resonance splitting induced by sidewall roughness in rough-
walled ring resonator OADMs.
 A comparison of the research findings with the already existing results are listed and
summarized as below:
1. Series coupled ring resonator OADM:
i. Based on [24, 25], the critical values for inter-ring coupling coefficients that result
in a deep notch of the through port response of the filter were calculated. Above the
critical values, the response will show a resonance splitting.
ii. In CHAPTER Four, exploiting resonance splitting, and keeping the crosstalk
suppression over a sufficient level (|20|𝑑𝐵) yielded a design of an OADM with a
crosstalk suppression bandwidth sufficient to drop 10 Gbps (RZ and NRZ) with a
mitigated level of crosstalk. The over-coupling condition between rings to produce
wider crosstalk suppression was proposed.
147
iii. A comparison between the over-coupled design suggested in this thesis with the
critical coupled series ring resonator [24] showed an increase of about 40% in
crosstalk suppression bandwidth for similar outer coupling coefficients (similar gap
width).
2. Parallel coupled ring resonator OADM:
i. Based on [81, 102], long and sophisticated analytical forms of the transfer function
(using scattering method) were proposed. These forms are hard to use for crosstalk
calculations.
ii. In CHAPTER Five, the transfer function was derived analytically using the Signal
Flow Graph method (based on Mason’s rule), for the first time. A closed form of the
spectral response was presented. It provides a simple and direct method to calculate
the transfer function of the parallel coupled OADM.
iii. The derived transfer function was used to calculate the crosstalk suppression
bandwidth. This model provides the possibility of examining different parameters
that affect the crosstalk and OBRR levels.
3. Vertically coupled single ring OADM:
i. Based on [23, 77], vertical coupled ring resonator OADMs were used to improve
crosstalk performance by increasing the crosstalk suppression ratio.
ii. In CHAPTER Six, an electromagnetic simulation-driven design optimization
method procedure was used to optimize the ring parameters (vertical separation,
lateral deviation, and waveguide’s height) in order to increase the crosstalk
148
iii. The pattern search algorithm was used and a 21 GHz crosstalk suppression
bandwidth was shown to be achievable by arranging the height of the waveguide as
well as the vertical and lateral separation. This bandwidth is sufficient to drop a 10
Gbps NRZ transmission using a small size OADM.
4. Grating-assisted ring resonator OADM:
i. Based on [118, 162], controllable reflectivity resulting from a semi-periodic grating
was used in all-pass filters to improve the spectral response.
ii. In CHAPTER Seven, the resonance splitting induced by sidewall roughness was
exploited. The time and space domain CMT were used to derive the analytical
models that calculate the rough-walled ring parameters from experimental and
numerical results, without the need for curve fitting calculations. An optimization
technique based on the goal maximization algorithm (in Excel) was used to
calculate the back reflection coefficient that results in a wide bandwidth (over 20
GHz).
iii. A design of single ring OADM, with three gratings, that provides a 28 GHz
crosstalk suppression bandwidth was proposed.
 As a conclusion, a number of methods that can be applied to the existing OADM
designs, to improve the optical EMC, were proposed and validated. Several small-size
OADM designs, with a crosstalk suppression bandwidth > 20 𝐺𝐻𝑧, were presented and
optimized. These designs provide efficient dropping for 10 Gbps modulated channels in
WDM networks. Improving signal integrity in small size filters has the advantage of
149
enhancing the overall optical EMC in the PLCs and allows for increasing integration
density. The proposed designs can be used as a basis for higher order OADMs to improve
the response shape, and moreover, to support higher data rate transmissions (40 Gbps, 100
Gbps and 400 Gbps).
 In summary, issues of crosstalk have long been research topics in the Photonic
community. However, the term “Optical EMC” is becoming more widespread as the
reduction in wavelengths and the increase in scale of integration results in these phenomena
being increasingly barriers to successful operation. This thesis has approached the topic of
‘Photonics’ crosstalk with EMC approach. It is anticipated that this thesis will be a starting
point for future research rather than the last word on crosstalk in OADMs.
8.2. Suggestions for Future Work
EMC issues in optical systems are worthy of consideration. Therefore, several further
investigations would be beneficial for a fuller understanding of optical EMC. This thesis
addressed the crosstalk in small-sized ring resonator based OADMs. However, in order to
have further improvement in optical EMC in WDM networks, a number of directions can
be suggested for future work as listed below:
1. The influence of parasitic factors like unbounded (substrate) radiation, which could be
important in the optical EMC context, need to be considered. Generic foundry models [169]
and the related S-parameter models can be enhanced to simulate the unintended coupling to
adjacent components and stray light propagating modes in the design stage.
150
2. Two stage OADMs need to be analysed and optimised in terms of crosstalk suppression
bandwidth. Although a single stage module for all the operations is desirable, usually in
order to minimize the crosstalk effects a single stage is used to realize the drop function and
another single stage is used for the add function. Thus, the optimization of multi-stage
filters (e.g. high drop port rejection ratio, box-shaped response, etc.) is desirable.
3. The proposed structure in Chapter seven consists of a single ring resonator with partial
reflectors embedded in the ring. Further study of the counter-directional coupling by using
FBG on the coupling region would be beneficial to propose OADMs with improved
crosstalk performance.
4. Concentric rings based OADMs also deserves study. An increase in the through port
notch depth has been reported by placing another ring inside the resonator in all-pass filter
structure. Therefore, applying this for an OADM structure would increase the crosstalk
151
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APPENDICES
Appendix A: SFG METHOD FOR PARALLEL COUPLED OADMS
Filter characteristics such as higher selectivity and better Out-of-Band Rejection Ratio
(OBRR) can be obtained by increasing the order of the filter (𝑁 > 2). Mason’s rule is
applicable to any value of 𝑁. For 𝑁 rings in parallel, the number of forward paths is equal
to 𝑁, while the number of closed loops is ∑𝑁

𝑖=1 𝑖 . The general form of the transfer function
is:
𝐵(𝑋) (A -1)
𝐺=
𝐴(𝑋)
Where,
𝑁 (A-2)
𝐴(𝑋) = 1 + 𝑥1 ∑ [𝑎𝑚 ]𝑥2 2×𝑚
𝑚=0
𝑁−1 𝑁−𝑛
+ 𝑥1 2 ∑ [𝑎𝑚2 ]𝑥2 2×𝑚2 + ⋯ + 𝑥1 𝑛 . ∑ [𝑎𝑚𝑛 ] ⋅ 𝑥2 2×𝑚𝑛

𝑚2=0 𝑚𝑛=0
𝑛 = 3, … , 𝑁
𝑁 𝑁 (A-3)
1/2 𝑚−1 3/2
𝐵(𝑋) = 𝑥1 ∑ 𝑏1𝑚 . 𝑥2 + 𝑥1 ∑ 𝑏2𝑚 𝑥2 𝑚−1
𝑚=1 𝑚=1
𝑁
𝑁+1
+ ⋯ +𝑥1 2 ∑ 𝑏𝑛𝑚 𝑥2 𝑚−1
𝑚=1
After finding the general form, consider the case of two ring resonators, N=2.
173
𝑏11 𝑥11/2 + 𝑏12 𝑥11/2 𝑥2 2 + 𝑏21 𝑥1 3/2 + 𝑏22 𝑥1 3/2 𝑥2 2
𝐺=
1 + 𝑎11 𝑥1 + 𝑎12 𝑥1 𝑥2 2 + 𝑎2 𝑥1 2
((A.4)
2
𝑏11 = −𝑘1
𝑏12 = −𝑡1 2 𝑘2 2
𝑏21 = 𝑡2 2 𝑘1 2
𝑏22 = 𝑘1 4 𝑘2 2 + 𝑡1 4 𝑘2 2
𝑎11 = −(𝑡1 2 + 𝑡2 2 )
𝑎12 = −𝑘1 2 𝑘2 2
𝑎22 = 𝑡1 2 𝑡2 2
The intensity response is obtained by multiplying 𝐺 by 𝐺 ∗ , and the result is shown in (A.5)
𝑀 (A.5)
|𝐺|2 =
𝐷
𝑀 = 𝐵 + 𝐵3 𝑐𝑜𝑠𝛽𝐿𝑟 + 𝐵4 𝑐𝑜𝑠2𝛽𝐿𝑐 + 𝐵5 𝑐𝑜𝑠(𝛽𝐿𝑟 + 2𝛽𝐿𝑐 ) + 𝐵6 𝑐𝑜𝑠(𝛽𝐿𝑟 − 2𝛽𝐿𝑐 )
𝐷 = 𝐴1 + 𝐴2 𝑐𝑜𝑠𝛽𝐿𝑟 + 𝐴3 𝑐𝑜𝑠2𝛽𝐿𝑟 + 𝐴4 𝑐𝑜𝑠2𝛽𝐿𝑐 + 𝐴5 𝑐𝑜𝑠(𝛽𝐿𝑟 + 2𝛽𝐿𝑐 ) +
𝐴6 𝑐𝑜𝑠(𝛽𝐿𝑟 − 2𝛽𝐿𝑐 )
Where,
𝐵 = (𝑏11 2 + 𝑏12 2 )𝑒 −𝛼𝐿𝑟 + (𝑏21 2 + 𝑏22 2 )𝑒 −3𝛼𝐿𝑟
𝐵3 = (2𝑏11 𝑏21 + 2𝑏12 𝑏22 ) 𝑒 −2𝛼𝐿𝑟
APPENDICES 174
𝐵4 = (2𝑏11 𝑏21 𝑒 −𝛼𝐿𝑟 + 2𝑏12 𝑏22 𝑒 −3𝛼𝐿𝑟 )
𝐵5 = 2𝑏11 𝑏21 𝑒 −2𝛼𝐿𝑟
𝐵6 = 2𝑏12 𝑏22 𝑒 −2𝛼𝐿𝑟
And,
𝐴1 = 1 + 𝑎12 2 𝑒 −2𝛼𝐿𝑟 + 𝑎11 2 𝑒 −2𝛼𝐿𝑟 + 𝑎2 2 𝑒 −4𝛼𝐿𝑟
𝐴2 = 2𝑎11 𝑒 −𝛼𝐿𝑟 + 2𝑎11 𝑎2 𝑒 −3𝛼𝐿𝑟
𝐴3 = 2𝑎2 𝑒 −2𝛼𝐿𝑟
𝐴4 = 2𝑎11 𝑎12 𝑒 −2𝛼𝐿𝑟
𝐴5 = 2𝑎12 𝑒 −𝛼𝐿𝑟
𝐴6 = 2𝑎12 𝑎2 𝑒 −3𝛼𝐿𝑟
The phase constant is 𝛽 = 2𝜋𝑛𝑒𝑓𝑓 , and the condition that the phase constants for the bus
and bent waveguide sections are the same is considered. However, there is a small
difference between the effective refractive indices of straight and curved waveguides since
the field in the bent waveguide tends to propagate near the outer wall (rather than the centre
as in the bus waveguide) which means a lower velocity and higher effective refractive
index than in the bent waveguide; this is a relatively small difference [130].
APPENDICES 175
Appendix B: RUBY CODE FOR SIDEWALL ROUGHNESS GENERATION
#!/usr/local/bin/ruby -w
#
# Program to generate a collection of roughness voxels to add
# to a ring resonator. It should produce a .OBJ file for import
# into CST.
# 2 dimensional point
class BiPoint
attr_accessor :x, :y
include Comparable
def <=>(other)
[self.x, self.y] <=> [other.x, other.y]
end
def initialize(x,y)
@x, @y = x,y
end
# city block (Manhattan) distance function

def manhattan(other)
distance = (other.x - @x).abs + (other.y - @y).abs
return distance
end
# Check points are the same (x,y)

def ==(other)
return ((other.x == @x) and (other.y == @y))
end
# Vector addition
def +(other)
return BiPoint.new(@x + other.x, @y + other.y)
end
# reduce vector by scale factor. Raises

# ZeroDivisionError if k is zero.
def /(k)
raise ZeroDivisionError if k.zero?
return BiPoint.new(@x / k, @y / k)
end
end
# 3 dimensional point
class TriPoint
attr_accessor :x, :y, :z
APPENDICES 176
include Comparable
def <=>(other)
[self.x, self.y, self.z] <=> [other.x, other.y, other.z]
end
def initialize(x,y,z)
@x, @y, @z = x,y,z
end
# City Block distance but in 3-space.

# Cheaper to compute than Euclidean, but of
# sufficient utility.
def manhattan(other)
distance = (other.x - @x).abs + (other.y - @y).abs + (other.z -
@z).abs
return distance
end
# Map distance. There doesn't seem to be a good

# single word for horizontal distance in English.
# (Can't think of one in Esperanto either.)
# Can't really call it map_manhatten because ruby
# uses map as in {map, filter, reduce}.
def horizontal_manhattan(other)
distance = (other.x - @x).abs + (other.y - @y).abs
return distance
end
# vertical distance.
def vertical_manhattan(other)
distance = (other.z - @z).abs
return distance
end
# Check if they are the same point.

# Note, if they are floats the value of this may be
# almost nil.
def ==(other)
return ((other.x == @x) and (other.y == @y) and (other.z = @z))
end
# Add two vectors together

def +(other)
return TriPoint.new(@x + other.x, @y + other.y, @z + other.z)
end
# Reduce vector by a scale factor.

def /(k)
raise ZeroDivisionError if k.zero?
return TriPoint.new(@x / k, @y / k, @z / k)
end
APPENDICES 177
def to_vertex
"v #{@x.to_f} #{@y.to_f} #{@z.to_f}"
end
end
class Grid
VERBOSITY = 5 # controls diagnostic output ...

@@verbose = :verbose
def Grid.verbose(value)
@@verbose = value
end
def Grid.verbose?
@@verbose == :verbose
end
def verbose?
Grid.verbose?
end
# convert from the real space to grid co-ordinates.

# which are integers.
#
def scale_to_grid(x,y)
c = (((x - @xmin)/@step) + 0.5).to_i
r = (((y - @ymin)/@step) + 0.5).to_i
return r, c
end
# Set up a grid between (minx,miny) and (maxx,maxy)

# spaced (equally in both directions) by step
def initialize(minx, miny, maxx, maxy, step)
# The grid is actually a collection of
# TriPoints.
@xmin = minx
@ymin = miny
@xmax = maxx
@ymax = maxy
@step = step
@grid = []
Logger.logwrite "@step = #{@step}" if self.verbose?
y = @ymin
@rows = 0
while (y <= @ymax)
Logger.logwrite "initialize: y=#{y}" if self.verbose?
@rows += 1
@columns = 0
APPENDICES 178
x = @xmin
while (x <= @xmax)
if @@verbose == :verbose
if VERBOSITY == 10
Logger.logwrite "initialize: (xmin,ymin) <= (x, y) <=
(xmax,ymax)"
Logger.logwrite "............ (#{@xmin},#{@ymin} <= (#{x},
#{y}) <= (#{@xmax},#{@ymax})"
Logger.logwrite "@step = #{@step}"
end
end
@columns += 1
@grid << TriPoint.new(x,y,0)
x += @step
end
y += @step
end
end
# Populate the grid at k with v. If there is already

# data there it is added to unless op is :replace, or
# something else. Something else should use method missing.
# Probably on TriPoint.
def populate(k,v, op=:add)
Logger.logwrite "populate: k: #{k.inspect} => #{v}" if self.verbose?
r,c = scale_to_grid(k.x, k.y)
Logger.logwrite "(c,r) is (#{c},#{r})" if self.verbose?
irc = index(r,c)
Logger.logwrite "@grid[#{irc}] = #{@grid[irc]}" if self.verbose?
p0 = BiPoint.new(@grid[irc].x,
@grid[irc].y)
# Check we are looking at the right place
if (p0.manhattan(k) < (2 * @step))
case op
when :add
#Just move the point up.
@grid[irc] += TriPoint.new(0,0,v)
when :replace
@grid[irc] = TriPoint.new(k.x,k.y,v)
else
@grid[irc].send(:op, k, v)
end
else
# Not looking in the right place
if @@verbose == :vebose
Logger.logwrite "KLAXON!!!"
Logger.logwrite "r = #{r}, c = #{c}, p0 = #{p0.inspect},"
Logger.logwrite "@grid[index(r,c)] =
#{@grid[index(r,c)].inspect}"
end
end
end
APPENDICES 179
# Given a row and column, find the entry in Grid, which
# is a normal array, not 2D
def index(r,c)
(r * @columns) + c
end
# In .OBJ format, vertices are numbered from 1, not zero.

def vertex_index(r,c)
index(r,c) + 1
end
# Make a Coarser grid, so that spikiness in the grid

# is smoothed out.
def subsample(k)
Grid.verbose(@@verbose)
subgrid = Grid.new(@xmin, @ymin, @xmax, @ymax, @step * k)
0.step(@rows,k) do |r|
0.step(@columns,k) do |c|
sum = 0
(0...k).each do |i|
(0...k).each do |j|
idx = index(r+i, c+j)
sum += @grid[idx].z if @grid[idx]
end
end
p = @grid[index(r,c)]
subgrid.populate(p, sum)
end
end
return subgrid
end
# Outputs the grid in .OBJ format.

def display_grid(outfile)
Logger.logwrite "@columns = #{@columns}"
Logger.logwrite "@rows = #{@rows}"
Logger.logwrite "product @rows * @columns = #{@rows * @columns}"
Logger.logwrite "\nEND OF CALCULATIONS\n"
end
open(outfile, "w") do |outf|
@grid.each do |p|
vertex = p.to_vertex
Logger.logwrite vertex if @@verbose == :verbose
outf.puts vertex
end
r = 0
while (r < (@rows-1))
c = 0
while (c < (@columns-1))
face = "f #{vertex_index(r,c)} #{vertex_index(r,c+1)}
#{vertex_index(r+1,c+1)} #{vertex_index(r+1,c)}"
APPENDICES 180
Logger.logwrite face if @@verbose == :verbose
outf.puts face
c += 1
end
r += 1
end
end # close file

end
end
# define a rough surface over a hollow cylinder.

class Roughness
# Points list for a cuboid.

# The front face is 1,2,3,4, the back face is
# 5,6,7,8. We want this in triangles for best(?)
# meshing -- programs like Anim8or, and other .OBJ
# loaders seem to require it.
# 7--6
# |\ |
# | \|
# 7--3--2--6--7
# |\ |\ |\ |\ |
# | \| \| \| \|
# 8--4--1--5--8
# |\ |
# | \|
# 8--5
#
CUBENET=[[1,2,3],
[1,3,4],
[2,6,7],
[2,7,3],
[4,3,7],
[4,7,8],
[5,6,2],
[5,2,1],
[8,7,6],
[8,6,5],
[5,1,4],
[5,4,8]]
#
# Setup parameters for the the surface.
# r0 = internal diamere of the cylinder.
# r1 = external diameter of the cylinder.
# h = height of the cylinder
# dr = max depth of the roughness to penetrate
# the surface of the cylinder.
def initialize(r0, r1, h1, h2, dr)
@r0 = r0
@r1 = r1
@h1 = h1
APPENDICES 181
@h2 = h2
@dr = dr
@vertex_list = []
@face_list = []
@cube_count = 0
end
# Convert polar to rectangular co-ordinates, return as array.

# We don't need the height for this purpose as we are assuming the
# ring is horizontal.
def rectangular(r, theta)
return r*Math.cos(theta), r*Math.sin(theta)
end
# setup probability of a suface voxel being roughened, and

# the number of samples per degree.
def set_roughness_params(density = 0.25,
angular_granularity = Math::PI/180.0,
vertical_granularity = 1.0 * (@h2 - @h1))
@density = density
@angular_granularity = angular_granularity
@vertical_granularity = vertical_granularity
end
# Create a list of voxels by iterating around and over the

# inner and outer vertical surfaces of the ring.
# For now we don't bother with the top and bottom walls of
# the guide.
def create_list_of_voxels
theta = 0
toggle = 0
while theta < 2.0 * Math::PI
h = @h1
while h < @h2
# p = Kernel.rand()
# if p < @density
if toggle == 1
create_inside_voxel(h,theta)
@cube_count += 1
end
# p = Kernel.rand()
# if p < @density
if toggle == 1
create_outside_voxel(h,theta)
@cube_count += 1
end
h += @vertical_granularity
end
toggle = (toggle + 1) % 2
theta += @angular_granularity
end
APPENDICES 182
end
# Create a voxel on the inside surface of the ring

def create_inside_voxel(h, theta)
p = Array.new(8)
x,y = rectangular(@r0,theta)
p[0] = TriPoint.new(x,y,h)
x,y = rectangular(@r0,theta + @angular_granularity)
p[2] = TriPoint.new(x,y,h+@vertical_granularity)
x,y = rectangular(@r0+@dr,theta)
x,y = rectangular(@r0+@dr,theta + @angular_granularity)
x,y = rectangular(@r0+@dr,theta + @angular_granularity)
x,y = rectangular(@r0+@dr,theta)
p.each do |point|
@vertex_list.push(point.to_vertex)
end
CUBENET.each do |row|
@face_list.push("f " + row.map{|x| x + (8 * @cube_count)}.join("
"))
end
end
# Create a voxel on the outside surface of the ring

def create_outside_voxel(h, theta)
p = Array.new(8)
x,y = rectangular(@r1-@dr,theta)
x,y = rectangular(@r1-@dr,theta + @angular_granularity)
x,y = rectangular(@r1-@dr,theta + @angular_granularity)
x,y = rectangular(@r1-@dr,theta)
p.each do |point|
APPENDICES 183
@vertex_list.push(point.to_vertex)
end
CUBENET.each do |row|
@face_list.push("f " + row.map{|x| x + (8 * @cube_count)}.join("
"))
end
end
def create_alias_wavefront_file(filename)
File.open(filename,"w") do |fp|
@vertex_list.each do |vertex|
fp.puts vertex
end
@face_list.each do |face|
fp.puts face
end
end
end
end
class Roughness2 < Roughness

# Points list for a cuboid with no ends -- part of a cylinder.
# The missing front face is 1,2,3,4, the missing back face is
# 7--6
# |\ |
# | \|
# 7--3--2--6..7
# |\ | |\ | .
# | \| | \| .
# 8--4--1--5..8
# |\ |
# | \|
# 8--5
#
MESHNET=[[2,6,7],
[2,7,3],
[4,3,7],
[4,7,8],
[5,6,2],
[5,2,1],
[5,1,4],
[5,4,8]]
# For wiring up the last points

# 6--7
# |\ |
# | \|
APPENDICES 184
# 6--3--2--7..6
# |\ | |\ | .
# | \| | \| .
# 5--4--1--8..5
# |\ |
# | \|
# 5--8
#
CROSSWIRE=[[2,7,6],
[2,6,3],
[4,3,6],
[4,6,5],
[8,7,2],
[8,2,1],
[8,1,4],
[8,4,5]]

super(r0, r1, h1, h2, dr)
end

vertical_granularity = 0.1 * (@h2 - @h1))
@density = density
end
theta = 0
fcount = 0
while theta < (2.0 * Math::PI - @angular_granularity)
x1,y1 = rectangular(@r0 + @dr * Kernel.rand(), theta)
x2,y2 = rectangular(@r1 - @dr * Kernel.rand(), theta)
@vertex_list.push(TriPoint.new(x1,y1,@h1).to_vertex)
fcount += 1
if fcount >= 2
@cube_count += 1
end
end
@cube_count.times do |count|
MESHNET.each do |row|
@face_list.push("f " + row.map{|x| x + (4 * count)} .join(" "))
end
end
#join front and back faces. Hope this works
# MESHNET.each do |row|
# @face_list.push("f " + row.map{|x| (x + (4 * @cube_count)) % (4*
(@cube_count+1)) + 1}.join(" "))
APPENDICES 185
# end
# Close off front face

@face_list.push("f 1 2 3")
# close off back face
s = "f "
[0,-1,-2].each do |count|
v = @vertex_list.size + count
s = s + v.to_s + " "
end
@face_list.push(s)
s = s + v.to_s + " "
end
@face_list.push(s)
puts "@face_list size is #{@face_list.size}"

puts "@vertex_list size is #{@vertex_list.size}"
puts "@cube_count is #{@cube_count}"
puts "4 * (@cube_count+1) is #{4 * (@cube_count+1)}"
end
end
class Roughness3 < Roughness

# Points list for a cuboid with no ends -- part of a cylinder.
# The missing front face is 1,2,3,4, the missing back face is
# 7--6
# |\ |
# | \|
# 7--3--2--6..7
# |\ | |\ | .
# | \| | \| .
# 8--4--1--5..8
# |\ |
# | \|
# 8--5
#
MESHNET=[[2,6,7],
[2,7,3],
[4,3,7],
[4,7,8],
[5,6,2],
[5,2,1],
[5,1,4],
[5,4,8]]
APPENDICES 186
# For wiring up the last points
# 6--7
# |\ |
# | \|
# 6--3--2--7..6
# |\ | |\ | .
# | \| | \| .
# 5--4--1--8..5
# |\ |
# | \|
# 5--8
#
CROSSWIRE=[[2,7,6],
[2,6,3],
[4,3,6],
[4,6,5],
[8,7,2],
[8,2,1],
[8,1,4],
[8,4,5]]

super(r0, r1, h1, h2, dr)
end

vertical_granularity = (@h2 - @h1))
@density = density
end
theta = 0
fcount = 0
toggle = 0
while theta < (2.0 * Math::PI - @angular_granularity)
x1,y1 = rectangular(@r0 + (@dr * (toggle >> 1)), theta)
x2,y2 = rectangular(@r1 - (@dr * (toggle >> 1)), theta)
fcount += 1
if fcount >= 2
@cube_count += 1
end
toggle = (toggle + 1) % 4
end
@cube_count.times do |count|
MESHNET.each do |row|
@face_list.push("f " + row.map{|x| x + (4 * count)} .join(" "))
APPENDICES 187
end
end
#join front and back faces. Hope this works
# MESHNET.each do |row|
# @face_list.push("f " + row.map{|x| (x + (4 * @cube_count)) % (4*
(@cube_count+1)) + 1}.join(" "))
# end
# Close off front face

# close off back face
s = "f "
s = s + v.to_s + " "
end
@face_list.push(s)
s = s + v.to_s + " "
end
@face_list.push(s)
puts "@face_list size is #{@face_list.size}"

puts "@vertex_list size is #{@vertex_list.size}"
puts "@cube_count is #{@cube_count}"
puts "4 * (@cube_count+1) is #{4 * (@cube_count+1)}"
end
end
class GridToObj
@@verbose = :verbose
@@scaling = :linear
# Set the verbose flag to the supplied symbol
def GridToObj.verbose(sym)
@@verbose = sym
end
def GridToObj.verbose?
@@verbose == :verbose
end
# Read the data in a YAML file, and output the

# grid of points as a .OBJ file.
def GridToObj.process(*args)
output_names = []
Logger.logwrite "called GridToObj.process(#{args.join(", ")})\n"
APPENDICES 188
end
args.flatten!
if args[0] == "--log"
@@scaling = :logarithmic
args.shift
end
args.each do |name|
# Having the count overcomplicates things. Just follow
# the Unix philosophy.
# count = 1
open(name, "r") do |fp|
Logger.logwrite "process: About to create grid\n" if @@verbose ==
:verbose
outname = name.sub(/.ya?ml$/, ".obj")
Logger.logwrite "process: Will create #{outname}\n" if @@verbose
== :verbose
YAML::load_documents(fp) do |yaml_data|
grid = nil
Logger.logwrite "yaml_data is \n#{yaml_data.inspect}" if
self.verbose?
grid = self.create_grid(yaml_data)
Logger.logwrite "About to write grid to #{outname}\n" if
self.verbose?
grid.display_grid(outname)
Logger.logwrite "grid created.\n" if self.verbose?
output_names << outname
# count += 1
end
end
end
return output_names
end
# Diagnostic routine for logging, principally.

def self.inspect_data(raw_data)
begin
Logger.logwrite "create_grid: raw_data.class
is:\n#{raw_data.class}\n" if self.verbose?
Logger.logwrite "create_grid: raw_data is:\n#{raw_data.inspect}\n"
if self.verbose?
rescue => e
Logger.logwrite "attempt to print raw_data.inspect failed with
#{e.message}"
raise
end
end
#
def self.create_grid(raw_data)
self.inspect_data(raw_data)
Logger.logwrite "create_grid: call DataSet.new()..." if self.verbose?
#Create the collection of points.
APPENDICES 189
data = DataSet.new()
Logger.logwrite "create_grid: verify data is rectangle"
data.check_is_rectangular(raw_data)
data.add_array_to_dataset(raw_data,@@scaling)
ysize = raw_data.size
xsize = raw_data[0].size
Logger.logwrite "create_grid: calling Grid.verbose()" if
self.verbose?
Grid.verbose(@@verbose)
Logger.logwrite "create_grid: calling Grid.new(...)" if self.verbose?
grid = Grid.new(0, 0, xsize-1, ysize-1, 1)
data.each do |point|
Logger.logwrite "create_grid: point is #{point}" if self.verbose?
grid.populate(BiPoint.new(point.x, point.y), point.z, :replace)
end
return grid
end
end
class Logger
@@logfile = nil
def self.logwrite(*args)
unless @@logfile.nil?
open(@@logfile, "a") do |fp|
fp.puts args
end
end
end
def self.set_logfile(file)
@@logfile = file
end
end
if __FILE__ == $0
# Dimensions from slotted_ring_post_cst_advice....

r = 5.0
s = 1.0
h = 1.0
hg = 0.25
wr = 0.45
ws = 0.0
r1 = r - ((1-s) * wr) - ws
r0 = r1 - (s * wr)
roughness = Roughness.new(r0, r1, h, h + hg, (s*wr)/20.0)

roughness.set_roughness_params
roughness.create_list_of_voxels
roughness.("roughness_object.OBJ")
roughness2 = Roughness2.new(r0, r1, h, h + hg, (s*wr)/20.0)
APPENDICES 190
roughness2.set_roughness_params
roughness2.create_list_of_voxels
roughness2.create_alias_wavefront_file("roughness_object2.OBJ")
roughness3 = Roughness3.new(r0, r1, h, h + hg, (s*wr)/20.0)

roughness3.set_roughness_params
roughness3.create_list_of_voxels
roughness3.create_alias_wavefront_file("roughness_object3.OBJ")
end
APPENDICES 191

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Crosstalk and Signal Integrity in Ring Resonator

Based Optical Add/Drop Multiplexers for

Riyadh Dakhil Mansoor , B.Eng, M.Sc, MIET.

School of Engineering and Sustainable Development

A thesis submitted in partial fulfilment of the requirements for the degree of

degree or qualification in any other university.

simulator for validation.

CHAPTER ONE INTRODUCTION ....................................................................... 1

1.2. All-Optical Networks 6

1.3. Aims and Objectives 9

1.4. New Contributions to Knowledge 10

1.5. Outline of the Thesis 11

CHAPTER TWO CROSSTALK IN ALL-OPTICAL NETWORKS .................. 12

2.2. Optical Crosstalk 14

2.2.1. Classification of Crosstalk ........................................................... 14

2.2.2. Modelling of Crosstalk ................................................................ 17

2.3. Crosstalk In Optical Add/Drop Multiplexers 19

2.3.1. Array Waveguide Grating Based OADM ................................... 20

2.3.2. Fibre Bragg Grating Based OADM ............................................. 21

2.3.3. Ring Resonator Based OADM .................................................... 23

2.4. Crosstalk Modelling in Ring Resonator Based OADMs 24

2.4.1. Inter-Band Crosstalk .................................................................... 24

2.4.2. Intra-Band Crosstalk .................................................................... 26

2.5. Mitigation of Crosstalk in Ring Resonator Based OADMs 30

CHAPTER THREE OPTICAL RING RESONATORS ........................................... 35

3.2. Ring Resonators 37

3.3. SOI Strip Waveguides 42

3.3.1. Dispersion, Effective Refractive Index and Group Index ........... 43

3.3.2. Directional Coupler ..................................................................... 44

3.4. CST Microwave Studio 47

3.5. Coupling in Ring Resonators 49

3.5.1. Lateral Coupling .......................................................................... 49

3.5.2. Vertical Coupling ........................................................................ 50

3.6. Cascaded Coupled Ring Resonators. 51

3.6.1. Series Coupling ........................................................................... 51

3.6.2. Parallel Coupling ......................................................................... 53

3.7. Optical Add/Drop Multiplexer 54

CHAPTER FOUR OVER-COUPLED RING RESONATOR BASED OADM .... 60

4.2. Crosstalk Bandwidth in a Single Ring Resonator 63

4.2.1. CST Simulation ........................................................................... 64

4.2.2. Analytical Calculations ............................................................... 65

4.3. Crosstalk Bandwidth in Double Ring Resonator 70

CHAPTER FIVE CROSSTALK BANDWIDTH ESTIMATION OF

5.2. Mason’s Rule for Parallel Coupled Ring Resonators 83

5.3. CST Simulation 91

5.4. Analytical and Simulation Results 94

CHAPTER SIX VERTICALLY COUPLED RING RESONATOR OADM ...... 101

6.1. Introduction 101

6.2. Crosstalk Suppression: Analytical and Simulation Model 103

6.2.1. Analytical Model ....................................................................... 103

6.2.2. CST Simulation ......................................................................... 106

6.3. Crosstalk Suppression Bandwidth 109

6.4. Optimization Method 110

vii List of Contents

6.5. Conclusion 116

CHAPTER SEVEN GRATING-ASSISTED RING RESONATOR OADM ........ 117

7.1. Introduction 117

7.2. Coupled Mode Analysis 120

7.2.1. Time Domain Analysis .............................................................. 120