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Spreading Codes for All-Optical
Code Division Multiple Access
Communication Systems
M. Ravi Kumar
Spreading Codes for All-Optical
Code Division Multiple Access
Communication Systems
Thesis submitted to
Indian Institute of Technology, Kharagpur
for the award of the degree
of
Doctor of Philosophy
by
M. Ravi Kumar
under the guidance of
Professor S. S. Pathak
G. S. Sanyal School of Telecommunications
Indian Institute of Technology, Kharagpur
[2011]
c 2011, M. Ravi Kumar All rights reserved.
Dedicated to
My Family
Certificate of Approval
Date: / /
Certified that the thesis entitled Spreading Codes for All-Optical Code
Division Multiple Access Communication Systems, submitted by M. Ravi
Kumar to Indian Institute of Technology, Kharagpur, for the award of the
degree of Doctor of Philosophy has been accepted by the external examiners
and that the student has successfully defended the thesis in the viva-voce exami-
nation held today.
Signature:
Name:
(Member of DSC)
Signature:
Name:
(Member of DSC)
Signature:
Name:
(Member of DSC)
Signature:
Name:
(Supervisor)
Signature:
Name:
(External Examiner)
Signature:
Name:
(Chairman)
CERTIFICATE
This is to certify that the thesis entitled Spreading Codes for All-Optical
Code Division Multiple Access Communication Systems, submitted by M.
Ravi Kumar to Indian Institute of Technology, Kharagpur, is a record of bona
fide research work carried out under my supervision and is worthy of consideration
for the award of the degree of Doctor of Philosophy of the Institute.
S. S. PATHAK
Professor
Department of Electronics and Electrical
Communication Engineering
Indian Institute of Technology, Kharagpur
India 721302.
Date:
DECLARATION
I certify that
a. the work contained in this thesis is original and has been done by me under the
guidance of my supervisor.
b. the work has not been submitted to any other Institute for any degree or
diploma.
c. I have followed the guidelines provided by the Institute in preparing the thesis.
d. I have conformed to the norms and guidelines given in the Ethical Code of
Conduct of the Institute.
e. whenever I have used materials (data, theoretical analysis, figures, and text)
from other sources, I have given due credit to them by citing them in the text
of the thesis and giving their details in the references. Further, I have taken
permission from the copyright owners of the sources, whenever necessary.
M. Ravi Kumar
ix
Acknowledgments
I would like to thank all people who have helped and inspired me in the research
contributing to this thesis.
I am especially grateful to my thesis supervisor, Prof. Sant Sharan Pathak,
for his invaluable guidance during my research, encouragement to explore parallel
paths and freedom to pursue my ideas. My association with him has been a great
learning experience. He made it possible for me to discuss with a number of people
and work in different areas.
I express my sincere gratitude to Emeritus Prof. N. B. Chakrabarti for all the
knowledge he shared with me, without which our research would have been difficult
to conclude. Many thanks to Dr. P. Ganguly for his collaboration in our research
on integrated-optics. Thanks are also due to members of the Doctoral Scrutiny
Committee viz. Prof. D. Datta, Prof. R. V. Rajakumar, Prof. S. Chakrabarti
and Prof. D. Sarkar for their valuable comments and suggestions during my oral
presentations. I wish to thank Dr. Jayashree Ratnam for the discussions we had.
My sincere thanks to my lab mates Lakshi, Jinesh, Sahu Sir, Preetam Sir,
Sanjeet, Janardan, Debarati Madam, Prasad, Aruna Madam, Ashraf, Uma Sir,
Muthu, Jaydeb, Parul, Patil Sir, Sanjay Sir, Seemanti, Soumendra, Anil, Praful,
Anil Sir, Rashmi and Subbarao for their help and support. I would like to thank
Mr. Arunava Chaudhuri, Mr. Robert Anthony and Mr. Munna Pathak for their
technical assistance in Networks lab and Digital Communication lab.
I am also thankful to my friends especially Deba, Manoj, Prasant, Subrata,
Falguni, Atal bhai and Malaya for all the fun filled moments with them. Special
thanks to Mrs. Subrata and Mrs. Manoj for welcoming me many times to enjoy
some delicious food. This thesis would not have been possible without the constant
support I got from my family.
Lastly, I would like to thank all the people who made my stay at IIT Kharagpur,
an enjoyable one.
M. Ravi Kumar
xi
List of Important Abbreviations
1D one-dimensional
2D two-dimensional
3D three-dimensional
AWG arrayed waveguide grating
BER bit error rate
BIBD balanced incomplete block design
BPM beam propagation method
CDMA code division multiple access
CDR clock and data recovery
CRWOP complete row-wise orthogonal pairs
DF difference family
DTMF dual-tone multi-frequency
E/D encoder/decoder
EIMM effective index based matrix method
EOE CDMA electrical - optical - electrical CDMA
FEC forward error correction
FOOC folded optical orthogonal code
GF(p) Galois field of a prime number p
GMWPC generalized multi-wavelength prime code
GMWRSC generalized multi-wavelength Reed-Solomon code
IM/DD Intensity modulation and direct detection
LAN local area network
MAI multiple access interference
MAN metropolitan area network
MPP Multiple pulse per plane
MPR Multiple pulse per row
MWOOC Multiwavelength OOC
OC optical code
xiii
List of Abbreviations
OCDMA optical code division multiple access
OCFHC one-coincidence frequency hop code
OOC optical orthogonal code
OTDMA optical time division multiple access
PIIN phase-induced intensity noise
PON passive optical network
QCCM quadratic congruence code matrices
RWOP row-wise orthogonal pairs
SCBIBD strictly T-cyclic balanced incomplete block design
SPECTS Spectral phase-encoded time-spreading
SPP single pulse per plane
SPR single pulse per row
SSFBG superstructured fiber Bragg gratings
TE transverse electric
Ti:LiNbO
3
Titanium indiffused Lithium Niobate
TM transverse magnetic
WDM wavelength divison multiplexing
WDMA wavelength division multiple access
ZDC zero-gap directional coupler
xiv
List of Important Symbols

T
modulo-T subtraction
β propagation constant
C
s
surface concentration of indiffused titanium
C(x, z) concentration of indiffused titanium
η excitation efficiency
η(2D) spectral efficiency of 2D optical code family
η(3D) spectral efficiency of 3D optical code family
K weight of a 1D optical code
K

weight of a 2D optical code
K
′′
weight of a 3D optical code
λ
a
autocorrelation constraint
λ
c
crosscorrelation constraint
l length of optical waveguide
L
c
critical coupling length
n
e
extra-ordinary refractive index
n
o
ordinary refractive index
N number of interfering codes
N
max
cardinality of an optical code
P
e
probability of error due to MAI of a 1D OCDMA code
family
P

e
probability of error due to MAI of a 2D OCDMA code
family
P
′′
e
probability of error due to MAI of a 3D OCDMA code
family
S number of space channels or fibers
τ thickness of deposited titanium
t diffusion time
T number of time chips or temporal length of an optical
code
xv
List of Symbols
t
i
i
th
time chip of an optical code
∆t time delay between two optical pulses
Υ diffusion temperature
U
i
i
th
user
w width of single-mode waveguide
w wavelength of operation (µm)
w
i
i
th
wavelength
W number of wavelengths
xvi
List of Figures
1.1 Application of a OCDMA systems in a fiber optic network . . . . . 4
2.1 An EOE CDMA network model . . . . . . . . . . . . . . . . . . . . 21
2.2 An all-optical CDMA network model . . . . . . . . . . . . . . . . . 21
2.3 Spreading of 1D time spread codes . . . . . . . . . . . . . . . . . . 23
2.4 Spreading of a 2D SPR code . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Spreading of a 2D MPR code . . . . . . . . . . . . . . . . . . . . . 23
2.6 Spreading of a 3D SPP space - wavelength - time spread code . . . 23
2.7 Spreading of a 3D MPP space - wavelength - time spread code . . . 23
2.8 An optical encoder to generate a time spread 1D OCDMA code . . 24
2.9 Waveforms showing the generation of a 1D OCDMA code . . . . . . 25
2.10 Optical decoding of a spread 1D OCDMA code . . . . . . . . . . . 25
2.11 Waveforms showing the autocorrelation function . . . . . . . . . . . 26
2.12 An optical encoder to generate a 2D MPR OCDMA code . . . . . . 27
3.1 Flowchart depicting the proposed RWOP algorithm . . . . . . . . . 52
3.2 Cardinality of 1D OOCs and 2D RWOP-based code families . . . . 59
3.3 Probability of error due to MAI of 2D RWOP-based code families . 59
3.4 Probability of error due to MAI of 2D RWOP-based code families . 61
3.5 Comparison of 2D RWOP-based, MWOOC and GMWRSC families 61
3.6 Validation of analytical results by simulation . . . . . . . . . . . . . 62
3.7 Cardinality of 1D OOCs and 3D RWOP-based code families . . . . 67
3.8 Spectral efficiency of proposed 2D and 3D code families . . . . . . . 67
3.9 Probability of error due to MAI of 3D RWOP-based code families . 69
3.10 Probability of error due to MAI of 3D RWOP-based code families . 69
3.11 Probability of error due to MAI of 3D RWOP-based code families . 69
3.12 Comparison of S < W and S = W 3D RWOP-based code families . 69
3.13 Comparison of SPP with 3D RWOP-based code families . . . . . . 70
3.14 Comparison of 2D and 3D RWOP-based code families . . . . . . . . 70
xvii
LIST OF FIGURES
4.1 Flowchart depicting the proposed CRWOP algorithm . . . . . . . . 76
4.2 Cardinality of 2D code families . . . . . . . . . . . . . . . . . . . . 80
4.3 Spectral efficiency of 2D code families . . . . . . . . . . . . . . . . . 80
4.4 Probability of error due to MAI of 2D CRWOP-based code families 82
4.5 Comparison of 2D CRWOP, RWOP and MWOOC families . . . . . 82
4.6 Comparison of MWOOC, RWOP and CRWOP 2D code families . . 83
4.7 Cardinality of CRWOP-based, RWOP-based and SPP 3D families . 88
4.8 Spectral efficiency of proposed 3D code families . . . . . . . . . . . 88
4.9 Probability of error due to MAI of CRWOP-based 3D families . . . 89
4.10 Comparison of CRWOP-based, RWOP-based and SPP 3D families . 89
4.11 Comparison of SPP, RWOP and CRWOP 3D code families . . . . . 90
5.1 A conventional integrated optic directional coupler . . . . . . . . . 96
5.2 Proposed integrated optic zero-gap directional coupler . . . . . . . . 96
5.3 Deposited Titanium strips on a LiNbO
3
substrate . . . . . . . . . . 97
5.4 Ordinary refractive index profiles . . . . . . . . . . . . . . . . . . . 98
5.5 Extra-ordinary refractive index profiles . . . . . . . . . . . . . . . . 98
5.6 Propagation constant versus excitation efficiency of TE modes . . . 100
5.7 Propagation constant versus excitation efficiency of TM modes . . . 100
5.8 Propagation length versus crosstalk . . . . . . . . . . . . . . . . . . 101
5.9 Miniature 1D OCDMA code generator using ZDC . . . . . . . . . . 103
5.10 Input and output pulses of MOCG:Z . . . . . . . . . . . . . . . . . 106
5.11 Input and output pulses using a polarizer . . . . . . . . . . . . . . . 106
5.12 Miniaturized 1D OCDMA code generator using a Y-junction . . . . 107
5.13 Delay between the output pulses . . . . . . . . . . . . . . . . . . . . 109
5.14 Miniature 2D OCDMA code generator using ZDCs . . . . . . . . . 110
5.15 Miniature 3D OCDMA code generator using ZDCs . . . . . . . . . 112
xviii
List of Tables
2.1 Difference table of a cyclic difference set . . . . . . . . . . . . . . . 30
2.2 Example: four user, K = 2 OOC family . . . . . . . . . . . . . . . . 35
3.1 Example 2D GMWPC construction for W = 4, T = 7 & K

= 4 . . 44
3.2 Example 2D GMWRSC construction for W = 3, T = 10 & K

= 2 . 46
3.3 Example 2D MWOOC construction for W = 3, T = 9 & K

= 2 . . 48
3.4 Example 2D FOOC construction for W = 3, T = 9 & K

= 4 . . . . 49
3.5 Example 2D code construction for W = 8, K

= 6 . . . . . . . . . . 57
3.6 Example 2D code construction for W = 8, K

= 4 . . . . . . . . . . 57
3.7 Example 2D code construction for W = 4, T = 7 & K

= 4 . . . . . 62
3.8 Comparison of example 2D code families . . . . . . . . . . . . . . . 63
3.9 Example 3D code construction for S = W = 8, K
′′
= 8 . . . . . . . 64
3.10 Example 3D code construction for S = 4, W = 8, K
′′
= 8 . . . . . . 65
4.1 Example of empty wavelength grid for W = 7 . . . . . . . . . . . . 76
4.2 Example of allocated 1D OOCs in wavelength grid for W = 7 . . . 76
4.3 Example of created array based on wavelength pairs for W = 7 . . . 77
4.4 Example of array sorted according to users’ 1D OOCs for W = 7 . . 77
4.5 Example of CRWOP generated array for W = 7 . . . . . . . . . . . 77
4.6 Example 2D code construction for W = 9, K

= 4 . . . . . . . . . . 79
4.7 Simpler form (same example as Table 4.6) . . . . . . . . . . . . . . 80
4.8 Example 2D code family for W = 5, T = 7 & K

= 4 . . . . . . . . 83
4.9 Comparison of RWOP-based and CRWOP-based 2D code families . 84
4.10 Example of 3D code construction (S = W = 7, K
′′
= 8) . . . . . . . 86
4.11 Example of S < W 3D code construction (S = 4, W = 5, K
′′
= 8) . 86
4.12 Codes of all users (same example as Table 4.11) . . . . . . . . . . . 87
xix
Abstract
In code division multiple access (CDMA) network design, the error due to multiple
access interference (MAI) is an important factor which is mitigated to some extent by
generating almost orthogonal codes for very small autocorrelation and crosscorrelation
values. The optical CDMA (OCDMA) design is based on binary, unipolar spreading
codes, which in turn requires considerably longer length of spreading codes in order
to satisfy these constraints. Spreading the optical spectrum in wavelength, time and
multiple fibers is observed to satisfy the constraints for accommodating sufficiently large
number of users with a comparatively smaller spreading code length. In all-optical
CDMA design, the information in electrical domain is spread by directly assigning a
pulse pattern in optical domain in order to accommodate data rate of the order of
Terabits per second. Generation of such patterns requires multiple number of delay
lines stacked in parallel. These delay lines, if obtained in the form of optical fiber of
variable length, are difficult to integrate along with laser technology in miniaturized
form which is expected to be a demand of network technology in future. In this thesis,
some aspects of these problems are considered.
A row-wise orthogonal pairs (RWOP) algorithm is proposed and analyzed for its appli-
cation to wavelength - time two-dimensional (2D) OCDMA code family construction.
The probability of error due to MAI of RWOP codes is smaller than that for other
2D constructions available in literature. The RWOP 2D codes are extended to space -
wavelength - time three-dimensional (3D) code families as well for its performance gain
which supports the backward compatibility to 2D like any other 2D, 3D code family
constructions. The number of codes using RWOP algorithm are further enhanced to
accommodate more number of users in 2D as well as 3D OCDMA. This enhancement
has been brought in with the help of another algorithm designed, developed and an-
alyzed in this thesis. This new algorithm is named as complete row-wise orthogonal
pairs (CRWOP). Further miniaturization of code generation procedure by integrating
laser technology and delay realization in integrated-optic form is considered next. Two
different configurations of integrated-optic devices for generating OCDMA codes are
explored. One is a new TE-TM mode splitter based on a zero-gap directional coupler
and the other is a 3dB power splitting Y-junction. The spreading of the optical pulses,
size, insertion loss and maximum number of users supported by the proposed 2D and 3D
codes using titanium indiffused lithium niobate integrated-optic technology is explored.
Key Words - OCDMA, Unipolar codes, MAI, Integrated-optic code generators
xxi
Contents
Title Page i
Certificate of Approval v
Certificate vii
Declaration ix
Acknowledgements xi
List of Abbreviations xii
List of Symbols xv
List of Figures xvii
List of Tables xix
Abstract xxi
1 Introduction and Review 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 One-Dimensional OCDMA Systems . . . . . . . . . . . . . . . . . . 3
1.3 Two-Dimensional OCDMA Systems . . . . . . . . . . . . . . . . . . 9
1.4 Three-Dimensional OCDMA Systems . . . . . . . . . . . . . . . . . 15
1.5 Contributions made in the Thesis . . . . . . . . . . . . . . . . . . . 16
1.6 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 17
xxiii
CONTENTS
2 OCDMA System Model 19
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Architecture of OCDMA Networks . . . . . . . . . . . . . . . . . . 20
2.2.1 Spreading Pattern of OCDMA Code Families . . . . . . . . 22
2.2.2 Generation of OCDMA Codes . . . . . . . . . . . . . . . . . 23
2.3 Construction of OCDMA Code Families . . . . . . . . . . . . . . . 28
2.3.1 Large Weight, Small Length Codes: Prime Sequences . . . . 28
2.3.2 Minimum Correlated OOCs . . . . . . . . . . . . . . . . . . 29
2.3.3 Optimal Length, Minimum Correlated OOCs . . . . . . . . . 31
2.3.4 Probability of Error Due to MAI . . . . . . . . . . . . . . . 33
2.4 Feasibility of Getting Large Delay in OCDMA Code Generation . . 36
2.5 Applications of Lithium Niobate Devices . . . . . . . . . . . . . . . 37
2.6 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . 38
3 OCDMA Code Families based on a Novel RWOP Algorithm 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Two-Dimensional OCDMA Code Families . . . . . . . . . . . . . . 42
3.2.1 GMWPCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.2 GMWRSCs . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.3 MWOOCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.4 FOOCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Three-Dimensional OCDMA Code Families . . . . . . . . . . . . . . 50
3.4 The RWOP Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5 RWOP-based 2D OCDMA Code Families . . . . . . . . . . . . . . . 55
3.5.1 Construction of RWOP-based 2D Code Families . . . . . . . 56
3.5.2 Analysis of RWOP-based 2D Code Families . . . . . . . . . 57
3.6 RWOP-based 3D OCDMA Code Families . . . . . . . . . . . . . . . 63
3.6.1 Construction of RWOP-based 3D Code Families . . . . . . . 63
3.6.2 Analysis of RWOP-based 3D Code Families . . . . . . . . . 65
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4 OCDMA Code Families based on a Novel CRWOP algorithm 73
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Scope of Improvement in RWOP Algorithm . . . . . . . . . . . . . 74
4.3 The CRWOP Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4 CRWOP-based 2D OCDMA Code Families . . . . . . . . . . . . . . 78
xxiv
CONTENTS
4.4.1 Construction of CRWOP-based 2D Code Families . . . . . . 78
4.4.2 Analysis of CRWOP-based 2D Code Families . . . . . . . . 79
4.5 CRWOP-based 3D OCDMA Code Families . . . . . . . . . . . . . . 84
4.5.1 Construction of CRWOP-based 3D Code Families . . . . . . 85
4.5.2 Analysis of CRWOP-based 3D Code Families . . . . . . . . 86
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5 Miniaturization of OCDMA Code Generation 93
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Zero-Gap Directional Coupler . . . . . . . . . . . . . . . . . . . . . 95
5.2.1 Determination of Critical Coupling Lengths . . . . . . . . . 96
5.2.2 Design of ZDC . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.3 Ti:LiNbO
3
based 1D OCDMA Code Generation . . . . . . . . . . . 102
5.3.1 ZDC Based Code Generation . . . . . . . . . . . . . . . . . 102
5.3.2 Y-junction Based Generation . . . . . . . . . . . . . . . . . 106
5.3.3 Delay Comparison . . . . . . . . . . . . . . . . . . . . . . . 108
5.4 Ti:LiNbO
3
based 2D OCDMA Code Generation . . . . . . . . . . . 110
5.4.1 Design Considerations . . . . . . . . . . . . . . . . . . . . . 110
5.4.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . 111
5.5 Ti:LiNbO
3
based 3D OCDMA Code Generation . . . . . . . . . . . 111
5.5.1 Design Considerations . . . . . . . . . . . . . . . . . . . . . 112
5.5.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . 112
5.6 Insertion Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6 Conclusions and Future Directions 117
6.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.1.1 2D & 3D RWOP-based OCDMA Code Families . . . . . . . 118
6.1.2 2D & 3D CRWOP-based OCDMA Code Families . . . . . . 118
6.1.3 Miniaturization of OCDMA Code Generation . . . . . . . . 119
6.2 Scope for Further Study . . . . . . . . . . . . . . . . . . . . . . . . 120
References 122
Publications from the Thesis Work 137
Curriculum Vitae 139
xxv
C H A P T E R 1
Introduction and Review
1.1 Introduction
The development of the economy and society needs improved performances of
communication networks, in particular optical communication networks. Optical
backbone networks have been able to afford the requirement of bandwidth, whereas
access networks are still the bottleneck and will be of most importance in network
design and construction. Optical code division multiple access (OCDMA) is one
of the emerging technologies for future multiple access networks along with wave-
length division multiple access (WDMA) [1] and optical time division multiple
access (OTDMA) [2]. The trend of increased research on OCDMA is accelerating
due to fiber penetration in the first mile and the establishment of passive optical
network (PON) technology as a pragmatic solution for residential access. The con-
cept of OCDMA is based on that of the widely used code division multiple access
(CDMA) technology in microwave wireless communications where users are as-
signed spreading codes or signature sequences. The concept of CDMA [3] involves
Introduction and Review
sharing of a common communication channel among multiple users. Similarly,
in OCDMA a common optical communication channel is shared among multiple
users. Networks employing OCDMA may be asynchronous in which users can
transmit their assigned codes at any instant of time, or synchronous in which
users transmit their assigned codes at previously allotted time periods.
According to [4], the use of multiple high-capacity fibers for communications
networking with optical correlation by fiber tapped delay lines provides speedy
and easy-to-implement decoders. Thus the individual user obtains a transparent
low-speed channel by code multiplexing. The reliability of this low-speed channel
can be enhanced by redundantly coding the patterns sent by the user, for which
the encoding and decoding processes can be performed electronically. The elec-
tronic encoding of data followed by optical encoding is simple to implement, highly
reliable at reasonable throughput, and provides asynchronous access with simple
protocol. Even though both synchronous OCDMA [5,6,7] as well as asynchronous
OCDMA [6, 8, 9, 10] have been reported, this thesis is limited to the asynchronous
case.
Spread spectrum CDMA allows asynchronous multiple access to a local area
network (LAN) with no waiting. The additional bandwidth required by spread
spectrum can be accommodated by using a fiber-optic channel and incoherent
optical signal processing. New CDMA sequences are designed specifically for op-
tical processing. Gold codes are not suitable for OCDMA systems because they
exhibit a large crosscorrelation variance [11]. In OCDMA, an optical code (OC)
represents a user address and signs each transmitted data bit. Optical coding is
the process by which a code is inscribed into, and extracted from, an optical sig-
nal. Although a prerequisite for OCDMA, optical coding boasts a wide range of
novel and promising applications, such as OC label switching. The advantages of
OCDMA are often specific to particular technologies, techniques, or components,
2
1.2 One-Dimensional OCDMA Systems
thus implying drawbacks and tradeoffs. For instance, the cheapness of incoherent
light sources implies a limit on the network reach (due to dispersion), bit-error
rate (BER), and user bandwidth [12].
Optical communication channels for OCDMA may be either wireless [13] or
fiber-optic. We have confined our research work to fiber-optic CDMA. The digital
technology available in the electrical domain enables the use of bipolar codes
for CDMA, which have good correlation properties with large number of users.
In the optical domain, bipolar codes are restricted to optical phase shift keyed
communications like phase encoded OCDMA [8, 14, 15]. But fiber-optic phase
communication is prone to error as optical fibers are unable to maintain the phase
of an optical signal. Intensity modulation and direct detection (IM/DD) [16]
is an established fiber-optic communication technology. The IM/DD technology
requires the use of unipolar codes to give low error OCDMA systems.
One of the ways in which an OCDMA system can be integrated into a fiber
optic network is shown in Fig. 1.1. Different optical LAN and/or metropolitan
area network (MAN) networks can be interconnected by using all-optical CDMA
technology. The optical LAN/MAN networks may be any of electrical - opti-
cal - electrical CDMA (EOE CDMA), WDMA or OTDMA or a hybrid EOE
CDMA+WDMA network.
1.2 One-Dimensional OCDMA Systems
The working of a basic OCDMA system is explained in [11]. In OCDMA, a mode-
locked laser produces a low duty cycle, high intensity pulse stream at the data
rate. This sequence of pulses is modulated by an optical gate, such as a directional
coupler switch, which is driven by the information waveform. Using single-mode
optical fiber delay lines, each short laser pulse generates the appropriate code
3
Introduction and Review
1
2
3
4
N
1
2
3
4
N
Fiber−optic Interexchange/
Long−haul Network
(All−optical CDMA)
2
3
Gateway
Central
Office
(EOE−CDMA/WDMA/OTDMA)
Fiber−optic LAN/MAN
N
1
Figure 1.1: Application of a OCDMA systems in a fiber optic network
sequence. The optical fiber delay lines are configured so that K taps can be
selected from any of T positions, according to the address of the desired receiver.
At the receiver, correlation is performed by optical fiber delay lines. In order to
reduce the bandwidth requirements of the detector, the narrow autocorrelation
peak is used to trigger a bistable or monostable optical switch, with decay time
equal to the bit width. The slowly decaying signal is detected and processed at
the rate of the original data. Increasing the number of chips per bit, by using
optical processing, allows an increase in capacity of an OCDMA LAN.
Among the many different types of code sequences reported, prime sequences
were first discussed for OCDMA having maximum crosscorrelation values greater
than 1 [17]. Kwong and Prucnal [18] have given the construction of prime codes
4
1.2 One-Dimensional OCDMA Systems
and the overall system for one-dimensional (1D) OCDMA including encoders and
decoders of the prime codes. Tunable prime code encoders and decoders for
OCDMA are reported in [19]. A special class of 2
n
code, derived from prime
sequences for OCDMA is constructed in [20] along with a hybrid coding architec-
ture comprising a modified tunable prime encoder and 2
n
decoder. A technique
for constructing codes is presented that provides a family of optimal optical or-
thogonal codes (OOCs) having correlation λ
a
= λ
c
= 2 [21]. The parameters
(T, K, λ
a
, λ
c
) are respectively (p
2m
−1, p
m
+1, 2, 2), where p is any prime and the
cardinality is p
m
−2.
Algebraically designed OOCs, named as extended quadratic congruence codes
having better correlation properties than prime sequences are reported in [22].
The algebraic construction of quadratic congruence codes for use in CDMA fiber-
optic LANs is reported [23]. For every odd prime p, (p − 1) codes exist. The
sequences are of length T = p
2
, and correlation properties are λ
a
≤ 2, λ
c
≤ 4.
Various techniques for algebraically constructing 2
n
codes [20] of weight 4 are
presented in [24]. The upper bounds on the cardinalities of these codes are also
provided. Although, the cardinalities of the codes are not as good as that of the
non-symmetric codes, it is important to note that the coding architectures used
by those non-symmetric codes create so severe power loss that OCDMA systems
are not feasible no matter how good the cardinalities of those codes are. A general
theorem on the cardinality of the 2
n
prime-sequence codes is provided in [25].
These codes possess the algebraic properties of both prime-sequence and 2
n
codes.
Optical encoding and decoding structures to optimize the system parameters of
these OCDMA networks are described.
Code families for OCDMA having low autocorrelation and crosscorrelation
values are termed as OOCs [22, 26, 27, 28, 29]. The fundamental principles and
systems performance analysis of OCDMA are given in [30, 31]. An OOC intended
5
Introduction and Review
for OCDMA is a collection of (0,1) sequences with good correlation properties,
i.e., the autocorrelation of each sequence exhibits the thumbtack shape and the
crosscorrelation between any two sequences remains low throughout. The use
of OOCs enables a large number of asynchronous users to transmit information
efficiently and reliably. The thumbtack-shaped autocorrelation facilitates the de-
tection of the desired signal, and low-profiled crosscorrelation reduces interference
from unwanted signals. Methodologies in the design and analysis of OOCs with
tools from projective geometry, the greedy algorithm, iterative constructions, al-
gebraic coding theory, block design, and various other combinatorial disciplines
are discussed in [26]. An algorithm based on the extended set concept enables
the design of OCDMA codes with best achievable correlation properties [27]. Two
systematic OOC design techniques based on extended sets are presented in [32].
The first technique is a deterministic design approach where the OOC sequences
are generated in a single run resulting in sequences of relatively short length. The
second technique is semi-random and may require multiple iterations until all OOC
sequences are generated converging to the optimal OOC.
Bounds on the size of OOCs with unequal autocorrelation and crosscorre-
lation values are developed and construction techniques for building them are
demonstrated in [33]. From results, an increase in the code size is possible by
letting the autocorrelation value exceed the crosscorrelation value. Among the
constructions given, the (T, K, 2, 1) codes are near-optimal, their cardinality is
N
max
= 2(T − 1)/K
2
and it is impossible to get more than 2(T − 1)/(K
2
− K)
codewords. Upper bounds on the size of an OOC are discussed in [34]. Sev-
eral constructions for optimal OOCs with weight 4 and correlation constraints
λ
a
= λ
c
= 1 are described by means of optimal cyclic packings. An equivalence
between optimal OOCs and optimal cyclic t-packings is established, whch allows
construction of optimal OOCs by way of optimal cyclic t-packings.
6
1.2 One-Dimensional OCDMA Systems
Some combinatorial constructions for optimal OOCs having λ
a
= λ
c
≤ 1 are
given in [28, 35, 36, 37]. The known techniques for constructing constant weight
codes are surveyed, and a table of (unrestricted) binary codes of length T ≤ 28 is
given in [38]. Three constructions for OCDMA code families having cyclic constant
weight are presented in [39]. All code families are asymptotically optimum, which
in turn means that, as the length of the sequences within the family approaches
infinity, the ratio of family size to the maximum possible under the Johnson up-
per bound approaches unity. A recursive construction for (T, K, λ
a
, λ
c
) OOCs is
presented in [40]. For the case of λ
a
= λ
c
, the recursive construction enlarges the
original family with λ
c
unchanged, and produces a family of asymptotically opti-
mal codes [39], if the original family is optimal. Some combinatorial constuctions
of optimal OOCs having λ
a
= λ
c
≤ 2 are given in [41, 42].
Constructions of difference families applicaple in 1D OCDMA are given in
[43,44]. Constructions of optimal (T, 4, 1, 1) OOCs [45] are shown by using perfect
difference families and cyclic pairwise balanced designs. A construction of OOCs
which is a generalization of the well-known construction of distinct difference set
by Bose and Chowla is provided in [29]. The construction is optimal with respect
to the Johnson bound and has parameters T = q
a
−1, K = q, and λ
a
= λ
c
= 1.
The application of an optical hard limiter in an OCDMA receiver to reduce the
effect of MAI is explained in [31, 46, 47]. Performance of asynchronous OCDMA
systems with double optical hard-limiters using OOCs is analyzed under the as-
sumption of Poisson shot noise model for the receiver photodetector where the
noise due to the detector dark currents exists [48]. Performance analysis of
OCMDA systems using OOCs and considering all major noise sources, i.e., quan-
tum shot-noise, dark current noise, and Gaussian circuit noise is discussed in [49].
Optical hard-limiters and high-speed integrate and dump circuits are two impor-
tant factors which make power efficient fiber-optic CDMA receivers realizable.
7
Introduction and Review
Asynchronous OCDMA systems with double optical hard-limiters have good per-
formance even when the number of simultaneous users is large. The exact bit
error probability of OCDMA systems employing optical hard limiters [47] is found
to be the generalization of the analysis in [46]. For λ
c
= 1 codes, the result is
not restricted to the case for threshold Th = K. The parallel interference can-
cellation (PIC) technique [50] can be used to remove MAI. Four direct sequence
OCDMA receivers based on the PIC technique with hard limiters placed before
the nondesired users or before the desired user receiver, or both are studied. For
the ideal synchronous case, the theoretical upper bound of the error probability
for the four receivers is given. The demonstration of synchronous OCDMA is re-
ported in [5]. A comparison of asynchronous and synchronous OCDMA based on
cardinality using prime codes is reported in [6]. Synchronous OCDMA systems
are also reported in [8, 9, 10, 51, 52, 53].
An optical AND logic gate receiver [54], which, in an ideal case, e.g., in the
absence of any noise source, except the optical MAI, is optimum. Direct and exact
solutions for OOCs with λ
c
= 1, 2, 3, . . . K, with the optical AND logic gate as
receiver are given. In most practical cases, OOCs with λ
c
= 2, 3 perform better
than OOCs with λ
c
= 1, while having a much bigger cardinality. An arrayed
waveguide grating (AWG)-based multiport optical encoder/decoder (E/D) and
forward error correction (FEC) technique are applied in an OCDMA system [55].
The AWG-based OCDMA E/D with high power contrast ratio between autocorre-
lation and crosscorrelation values can significantly suppress the interference noises
in an asynchronous OCDMA system without using ultra-long optical codes and
optical thresholder.
A limitation of 1D OOCs is that the length of the sequence increases rapidly
when the number of users or the weight of the code is increased, which means
large bandwidth expansion is required if a large number of codewords is needed.
8
1.3 Two-Dimensional OCDMA Systems
This can be overcome by the use of larger bandwidth in terms of using additional
dimensions to spread the codewords as discussed below.
1.3 Two-Dimensional OCDMA Systems
Two-dimensional OCDMA systems can be those which use any two domains
among space, wavelength, polarization, time and phase for spreading. This thesis
concentrates mainly on wavelength - time systems with IM/DD fiber optic com-
munication technology. A 2D OOC family is a set of W ×T matrices with (0, 1)
elements having low autocorrelation and crosscorrelation values. A brief review of
phase based 2D systems concludes this section.
An architecture for code-empowered OCDMA lightwave networks, based on
reconfigurable optically transparent paths among users of the network to provide
high-bandwidth optical connections on demand over small areas such as LANs or
access networks is presented in [56]. The network operates on the transmission of
incoherent OCDMA codes, each network station being equipped with an OCDMA
encoder and decoder. The routing at a network node is based on the OCDMA
code itself. The destination address, as well as the next node on the path, is
given by the code as in a code-empowered network. Commonly available delay
lines enable the tunability of the encoder, decoder, and router for a reconfigurable
and flexible network. A power analysis and focus on the performance issues of
dynamic routing is presented. The effect of coding, topology, load condition, and
traffic demand is analyzed using simulations. Routing rules, which are very unique
to OCDMA networks, are presented.
Multiwavelength OOCs (MWOOCs) [57] consist of 2D codewords with every
pulse of a codeword encoded in a distinct wavelength. Three classes of MWOOCs
based on OOCs, prime codes and Reed-Solomon codes have been constructed.
9
Introduction and Review
Using multiple wavelengths, the requirements of fiber ribbons and multiple stars
in space - time OCDMA networks are eliminated. A 2D OCDMA code family
having orthogonal properties in both the wavelength and time domains, which can
be physically implemented by using an array of Bragg gratings is reported in [58].
Multiple pulse per row (MPR) codes with optimum threshold detection maximizes
the cardinality and spectral efficiency [59]. A computationally efficient design
algorithm for MPR codes with optimum threshold detection is developed and a
simple receiver to enable real-time network optimization is shown. A construction
of (W×T, λ+2, λ
a
, λ
c
) [60] MWOOCs with the number of available wavelengths W,
codeword length T, and constant Hamming weight λ+2 that have autocorrelation
and crosscorrelation values not exceeding λ
a
= λ
c
is shown. There is no constraint
on the relationship between the number of available wavelengths and the codeword
length, and it is also possible to use an arbitrary λ
c
. The code is optimal for λ
c
= 1. The basic principles and the upper bound on the cardinality of a family of
wavelength-time MPR codes, for incoherent OCDMA networks, which have good
cardinality, spectral efficiency, and minimal crosscorrelation values are analyzed
in [61]. A family of one-coincidence frequency hop code (OCFHC)/OOC [62]
employs OCFHC and OOC as wavelength-hopping and time-spreading patterns
respectively. An algorithm to construct wavelength-time MPR codes, starting
with distinct 1D OOCs of a family as the row vectors of the code is given in [63].
In an OCDMA system using 2D single pulse per row (SPR) codes, a single
choice of the number of wavelength channels can accommodate different num-
bers of users with maximal spectral efficiency [64]. A fixed-hardware network
can readily be adapted in response to changes in the number of users and traffic
load by a readily scalable network or a time-dependent network. A family of 2D
wavelength-hopping time-spreading codes, which employs wavelength hopping al-
gebraically under prime-sequence permutations on top of time-spreading OOCs, is
10
1.3 Two-Dimensional OCDMA Systems
studied and analyzed [65]. These codes allow the number of wavelengths and code
length to be chosen independently and the code cardinality is a quadratic function
of the number of wavelengths without sacrificing the maximum crosscorrelation
value. A novel wavelength-aware detector for wavelength-hopping time-spreading
codes is discussed and shown to provide improved code performance. Wavelength
- time code families constructed with short code lengths [66] are given for OCDMA
networks. A family of 2D codes, having λ
a
= 0, λ
c
≤ 1, constructed by combin-
ing frequency-hop and time-spreading codes is presented in [67], which employ a
M-ary signalling scheme to increase the data transmission rate. A wavelength-
time coding scheme for high-speed OCDMA networks is reported in [68], and a
large number of new codes with asymptotically optimal cardinalities are gener-
ated by this coding scheme. This coding scheme has potential for applications of
secure optical networks. An incoherent OCDMA transceiver design, employing a
double-padded modified prime code family as spreading sequences, based on the
2D optical modulation scheme deploying frequency and polarization shift keying
is given in [69].
A 2D OOC construction scheme using the difference family (DF) is given in
[70]. The performance of the codes based on the received signal power is compared
with other codes. The system performance with double optical hard limiters by
using the Markov-chain method is also analyzed. The combinatorial properties
of 2D OOCs are revealed and an equivalent combinatorial description of a 2D
OOC is given in [71]. A special case of SCP called strictly T-cyclic balanced
incomplete block design (SCBIBD) is used to obtain optimal 2D OOCs. By using
(W×T, K

, 1)-SCBIBDs, new infinite classes of optimal (W ×T, K

, 1)-OOCs can
be obtained.
Flexibility of wavelength-time codes is investigated in [72], providing clarity on
the tradeoff between key code factors, specifically the number of available wave-
11
Introduction and Review
lengths and time chips. Since the number of available codes is always >> number
of active users at any given time for any truly asynchronous OCDMA system that
employs quasi-orthogonal codes, there is always a set of unused codes. These un-
used codes can be exploited to increase the spectral efficiency of the system [73] by
exclusively assigning to each user a set of M codes which represent a log
2
(M)-tuple
of bits so that each user effectively uses a multi-dimensional modulation (multiple
information bits per code are conveyed).
Replacing the SUM detector with the AND detector, the spectral efficiency
can be at least doubled with the same bandwidth, cardinality and probability of
error as reported in [74]. With respect to MAI, the AND detector is the optimum
single-user detector for any code with any dimension and weight. Practical clock
and data recovery (CDR) [75] for wavelength-time OCDMA provides an acceptable
BER penalty as compared to optimum sampling with a global clock. Results show
that MAI is not detrimental to practical CDR. A receiver without global clock [76]
providing quantization (to eliminate MAI), CDR, return-to-zero to non-return-to-
zero conversion (for OCDMA compatibility with digital logic), framing (for byte
synchronization), and FEC using a (255, 239) Reed-Solomon decoder, more than
doubles the number of supported users at a bit-error rate < 10
−10
. The receiver
supports an information rate of 156.25 Mb/s.
The temporal-spatial addition modulo T SPR codes having zero out-of-phase
autocorrelation and crosscorrelation value of one are constructed in [77]. A temporal-
spatial SPR prototype network with optical encoding and decoding using tapped
delay lines is built to test the autocorrelation and crosscorrelation properties of
these codes. The hard-limiting performance of 2D optical codes is analyzed under
the chip-asynchronous assumption in [78]. The experimental set-up of a three-
node 2D wavelength-time incoherent OCDMA system and study of output pulse
sequences of encoders and correlators for different number of active users and influ-
12
1.3 Two-Dimensional OCDMA Systems
ence of erbium doped fiber amplifier at input of correlators is demonstrated in [79].
The theoretical and experimental elimination of MAI in an incoherent OCDMA
system by using an incoherent dual code OCDMA receiver incorporating an ul-
trafast all-optical time gate is investigated in [80]. Experimental measurements
confirmed by simulations show error-free [BER < 10
−12
] operation for up to four
users. An integrated-photonic decoder [81] for 2D wavelength-time OCDMA, com-
posed of three AWGs, eight variable delay lines, and a 3 dB coupler is given. The
decoder utilizes complementary code processing and balanced detection to reduce
unwanted interference without using a threshold or time-gating device.
A wavelength-time OCDMA modulation scheme that does not use spread-
ing sequences for information transmission is presented in [82]. The proposed
transmitter sends coded data directly through the optical channel and exploits a
probabilistic method to reduce the amount of MAI. Simulations demonstrate that
the scheme can support large numbers of active users and is robust to the effects
of channel noise.
A space - wavelength OCDMA system [83] based on quadratic congruence
code matrices (QCCM) is given. According to the in-phase crosscorrelation of the
QCCM, MAI can be fully eliminated by using balanced photodetectors. Simula-
tions show efficient suppression of thermal noise, shot noise, and phase-induced
intensity noise (PIIN) in the receiver. A 2D space - wavelength code for spectral-
amplitude coding an OCDMA system is proposed in [84]. The corresponding E/D
pairs are based on the tunable fiber Bragg gratings cooperating with optical split-
ters/combiners. For the performance analysis, the effects of PIIN, shot noise, and
thermal noise are considered simultaneously. The BER performance compared
with that of the system using M−matrix codes allows larger number of active
users under a given BER.
Rapidly reconfigurable optical phase encoders and decoders based on fiber
13
Introduction and Review
Bragg gratings are shown in [85]. Spectral phase-encoded time-spreading (SPECTS)
OCDMA systems are described in [86, 87, 88, 89]. Spreading sequences for asyn-
chronous spectrally phase encoded OCDMA are given in [90]. Phase-wavelength
OCDMA systems are described in [8, 53, 91, 92, 93]. Encoders and decoders for
phase-wavelength OCDMA systems are shown in [14, 94]. The performance of
phase-wavelength OCDMA systems in dispersive fiber medium is analyzed in [95].
Coherent phase-wavelength coding techniques are given in [96]. Wavelength divi-
son multiplexing (WDM)-compatible spectrally phase-encoded OCDMA systems
are discussed in [97]. A full-duplex bidirectional spectrally interleaved OCDMA/dense
WDM system is described in [98].
Design, simulation, and experimental investigations of OCDMA networking
using SPECTS is discussed in [86]. Analysis has shown that nonuniform phase
coding can increase the orthogonality of the code set, thereby reducing the impact
of MAI. An experiment is conducted in a SPECTS OCDMA testbed incorporating
a highly nonlinear thresholder demonstrated error-free operation for four users at
1.25-Gb/s/user and for two users at 10-Gb/s/user. Walsh codes demonstrate
superior performance than m-sequences in the synchronous case, and the codes
achieve synchronous error-free operation at 1.25 Gb/s.
Asynchronous OCDMA over WDM systems have been experimentally demon-
strated using superstructured fiber Bragg gratings (SSFBG) and multi-port OCDMA
E/D in [99]. The total throughput is above 380 Gbit/s with a spectral efficiency
of about 0.32. Combined WDM/OCDMA systems using SSFBG are described
in [15, 100, 101].
14
1.4 Three-Dimensional OCDMA Systems
1.4 Three-Dimensional OCDMA Systems
Three-dimensional OCDMA systems can be those which use any three domains
among space, wavelength, polarization, time and phase for spreading. This thesis
concentrates mainly on space - wavelength - time systems with IM/DD fiber optic
communication technology. The 3D space - wavelength - time OCDMA code
families can be represented as (S ×W ×T, K
′′
, λ
a
, λ
c
).
A family of space - wavelength - time spread 3D optical codes for OCDMA
networks is reported in [102]. Codes with single pulse per plane (SPP) and multiple
pulses per plane (MPP), based on a prime sequence algorithm are shown. In
order to eliminate the requirement of fiber ribbons and multiple star couplers, a
wavelength
2
- time scheme has been suggested, in which the periodic property of
an AWG is used. For a small number of simultaneous users, the 3D MPP code
shows better performance due to dominant effect of increased threshold. The 3D
SPP code shows lower error probability for a large number of simultaneous users
since the effect of reduced crosscorrelation probability becomes dominant.
A 3D OCDMA transmission system that encodes data on time, wavelength
and polarization is experimentally demonstrated in [103]. This type of coding can
increase the cardinality by a factor of approximately 2
λc
over a conventional 2D
code. The performance of wavelength - polarization - time 3D OCDMA codes in
terms of bit error rate and Q factor with 1 Gbps, 1.5 Gbps, 2 Gbps, 2.5 Gbps, 3
Gbps,3.5 Gbps & 4 Gbps is studied in [104]. Such systems may be ideal for use
in short-distance optical LANs, where polarization states remain fairly stable.
Three dimensional perfect difference codes are constructed, and a correspond-
ing system structure for space - wavelength - time OCDMA is described in [105].
The codes, generated from the perfect difference set, can suppress the PIIN and
possess the MAI cancellation property. A family of 3D SPP codes for differ-
ential detection (SPDD) for OCDMA systems (based on the 1D golomb ruler
15
Introduction and Review
sequences), which achieve good cardinality and performance is presented in [106].
The improved BER performance is obtained by using two codes to encode ‘1’ and
‘0’ bits in the encoder and differential detection in the receiver. A family of 3D
space - wavelength - time codes for asynchronous OCDMA systems with off-peak
autocorrelation, λ
a
= 0, and peak crosscorrelation, λ
c
= 1, is reported in [107].
With W wavelengths and T time-slots, (W
2
T +W) codes are generated.
Though various types of 1D, 2D and 3D OCDMA code families and network
architectures are reported, a practically realisable system for fiber to the home
networks has not been reported so far. The thesis aims to bridge the gap between
the present and practically realisable OCDMA systems. The objective of the thesis
can be stated as:
• Design, development and testing of spreading codes for all-optical code di-
vision multiple access communication systems that possess the property of
low correlation, high cardinality and miniaturized generation.
1.5 Contributions made in the Thesis
The overall contributions of the thesis can be summarized below.
I. Construction, performance analysis and comparison of 2D MPR and 3D MPP
OCDMA code families based on a novel RWOP algorithm used for wavelength
and/or spatial allocation.
II. Construction, performance analysis and comparison of improved 2D MPR
and 3D MPP OCDMA code families based on a novel CRWOP algorithm
used for wavelength and/or spatial allocation.
III. Feasibility of lithium niobate based design of 1D, 2D and 3D miniature en-
coders for OCDMA code generation.
16
1.6 Organization of the Thesis
A brief description of the work carried out in the proposed thesis work is
presented below.
1.6 Organization of the Thesis
The functioning of different types of OCDMA networks is discussed in Chap-
ter 2. Some architectures of fiber based 1D, 2D and 3D OCDMA networks are
explained. The code families of such OCDMA systems given in literature are dis-
cussed. In Chapter 3, the construction and analysis of some of the existing 2D
and 3D OCDMA code families are discussed followed by the proposition of a row-
wise orthogonal pairs (RWOP) algorithm for wavelength and/or spatial allocation.
The RWOP algorithm is applied to construct 2D MPR and 3D MPP code families.
The performance of the constructed code families is analyzed and compared with
existing 2D and 3D code families. Chapter 4 deals with the construction and per-
formance analysis of 2D and 3D OCDMA code families based on a novel complete
row-wise orthogonal pairs (CRWOP) algorithm. The CRWOP algorithm is also
used for wavelength and/or spatial allocation. The construction of 2D MPR and
3D MPP code families is illustrated with the help of examples. The performance
of these CRWOP-based code families is compared with the RWOP-based code
families and other reported literature.
Based on some of the 1D, 2D and 3D OCDMA code families in Chapters 2,
3 and 4, miniaturization of 1D, 2D and 3D integrated-optic code generation is
considered in Chapter 5. The discussed integrated-optic devices are based on Ti-
tanium indiffused Lithium Niobate (Ti:LiNbO
3
) technology. Zero-gap directional
couplers (ZDCs) are designed as TE-TM mode splitters to be able to use the bire-
fringence property of LiNbO
3
. The simulation and design of a ZDC as a TE-TM
mode splitter is proposed. The application of the TE-TM mode splitter in de-
17
Introduction and Review
signing miniature 1D, 2D and 3D OCDMA code generators is worked out. The
designed miniature OCDMA code generators are compared with similar devices
in which the ZDC is replaced with a 3dB power splitting Y-junction. Finally, in
Chapter 6, the conclusions derived from this thesis are discussed and some aspects
of future directions of research are described.
18
C H A P T E R 2
OCDMA System Model
2.1 Introduction
An OCDMA network may be electrical - optical - electrical (EOE) or all - optical
in nature. In an OCDMA network, electrical data is communicated from the trans-
mitter end to the receiver end with an optical link in between. The transmitted
electrical data may be spreading codes in case of EOE CDMA, or raw data in case
of all - optical CDMA. When the transmitted electrical data are spreading codes,
they are converted into optical pulses by intensity modulation and fed to a channel
considered to be an optical fiber in this case, to be received by a photo-detector.
The output of the photo-detector contains the electrical spreading codes which
are decoded to get the data [108]. When the transmitted electrical data is raw,
encoding and decoding by spreading codes are done in the optical domain and is
termed as all - optical CDMA. All - optical network involves the communication of
optical data, eliminating the need for electrical to optical and optical to electrical
conversions [30]. In this chaper, the functioning of fiber based all - optical 1D,
OCDMA System Model
2D and 3D CDMA networks is explained. The different types and configurations
of spreading patterns for OCDMA code families used in 1D, 2D and 3D optical
networks are discussed. Typical constructions and properties of 1D OCDMA code
families are analyzed. The performance analysis of the reviewed 1D code families
in terms of probability of error due to multiple access interference (MAI), cardinal-
ity and spreading factor is explained. Some aspects of spreading and despreading
such OCDMA code families in all - optical domain are elaborated. Long length
all - optical CDMA code spreading is explored by different choice of materials.
Lithium niobate is observed to be a probable candidate for such applications on
the basis of the delay properties it exhibits.
2.2 Architecture of OCDMA Networks
The model of an EOE CDMA network as discussed in [4] is shown in Fig. 2.1. At
the transmitting end, raw electrical data is first encoded into CDMA sequences
using standard methods. The electrical CDMA sequence is then converted into
a sequence of optical pulses by direct or indirect modulation of a laser. The
sequences from different users are combined using an optical star coupler. The
star coupler distributes the combined code sequences to all the users. At the
receiving end, the combined code sequences are converted into electrical pulses
and the decoding of the CDMA sequence is done in the electrical domain.
The basic model of an all-optical CDMA network is shown in Fig. 2.2. A
similar model has been analysed in [26]. At the transmitter of each user, the net-
work converts raw electrical data into optical pulses either by direct or indirect
modulation of a laser. The optical pulses are encoded with a particular optical
code sequence by using an OCDMA encoder. The construction and design of an
OCDMA encoder varies depending on the optical code sequence to be encoded.
20
2.2 Architecture of OCDMA Networks
User 2
User N
User 2
User N
Electrical
CDMA Code
Laser
Optical
Pulses
User 1
Photo−
Detector
Electrical
CDMA Code
Star
Coupler
User 1
Figure 2.1: An EOE CDMA network model
The encoded sequences from different users are combined using a star coupler.
The star coupler distributes the combined code sequences to all the users. At the
receiver, the combined sequences are decoded using an OCDMA decoder corre-
sponding to the desired user. As in the case of OCDMA encoders, the structure
of OCDMA decoders also varies according to the optical code sequence to be de-
coded. A photo-detector converts the optical data back into electrical domain. It
may also be possible to use all-optical CDMA directly for optical data instead of
electrical data.
User 2
User N
Decoder
OCDMA
User 2
User N
Electrical
Data
Photo−
Detector
Electrical
Data
Encoder
Star
Coupler
User 1
OCDMA
User 1
Laser
Data
Optical
Figure 2.2: An all-optical CDMA network model
A set of codes constructed for a specific set of parameters is termed as a code
family. Unlike electrical CDMA, the spreading codes in OCDMA is of unipolar
type, which in turn are required to have a larger length for maintaining the im-
21
OCDMA System Model
pulsive autocorrelation property. To facilitate the design, the main parameters
involved in the construction of OCDMA code families are number of time chips
or temporal length (T), number of wavelengths (W), number of space channels
or fibers (S), weight (K, K

or K
′′
), maximum out-of-phase autocorrelation value

a
), maximum crosscorrelation value (λ
c
) and maximum number of users or car-
dinality (N
max
). We represent the weight of 1D, 2D and 3D codes as K, K

and
K
′′
respectively. Construction of these code families requires the autocorrelation
and crosscorrelation constraints to be satisfied. The autocorrelation constraint is
given as
T−1

t=0
x
t
x
t+˜ τ
≤ λ
a
for any code sequence x in a code family and any integer ˜ τ, 0 < ˜ τ < T. The
crosscorrelation constraint is given as
T−1

t=0
x
t
y
t+˜ τ
≤ λ
c
for any two code sequences x and y of a code family and any integer ˜ τ.
2.2.1 Spreading Pattern of OCDMA Code Families
A pictorial representation of the different types of codes is shown to understand
the 1D, 2D and 3D spreading patterns. Figure 2.3 shows two 1D codes spread in
the time domain, corresponding to users U
1
and U
2
. A 1D time spread OCDMA
code family is represented as (T, K, λ
a
, λ
c
), where T is the temporal length or
number of time chips in a code, K is the weight or the number of pulses in a code,
λ
a
is the maximum out-of-phase autocorrelation value and λ
c
is the maximum
crosscorrelation value. Figures 2.4 and 2.5 show wavelength - time spreading of a
2D SPR code and a 2D MPR code respectively. Similarly, a 2D wavelength - time
22
2.2 Architecture of OCDMA Networks
spread OCDMA code family is represented as (W ×T, K

, λ
a
, λ
c
), where W is the
number of wavelengths in the code family and K

is the weight of the 2D codes.
(1D) T →
U
1
:
U
2
:
Figure 2.3: Spreading of
1D time spread codes
(2D SPR) T →

W
Figure 2.4: Spreading of
a 2D SPR code
(2D MPR) T →

W
Figure 2.5: Spreading of
a 2D MPR code
Figures 2.6 and 2.7 show the spreading patterns of a 3D SPP and a 3D MPP
space - wavelength - time code respectively. Such a 3D OCDMA code family is
represented as (S ×W × T, K
′′
, λ
a
, λ
c
), where S is the number of space channels
or fibers used for the code family and K
′′
is the weight of the 3D codes.
(3D SPP) T →

W
s
i
(i
th
Fiber) s
j
(j
th
Fiber) s
k
(k
th
Fiber)
Figure 2.6: Spreading of a 3D SPP space - wavelength - time spread code
(3D SPP) T →

W
s
i
(i
th
Fiber) s
j
(j
th
Fiber) s
k
(k
th
Fiber)
Figure 2.7: Spreading of a 3D MPP space - wavelength - time spread code
2.2.2 Generation of OCDMA Codes
Among the different types of unipolar code families reported, most have λ
c
= 1
between two codes. Codes spread in the time domain are termed as 1D code
23
OCDMA System Model
families applicable in 1D OCDMA systems. The wavelength - time or space -
time spread codes are 2D and space - wavelength - time spread codes are 3D
codes, which are used in 2D and 3D OCDMA systems respectively. The concept
of space channels refers to fibers in a fiber-optic CDMA network, which means
that, each space channel points to a distinct fiber strand.
The block shown as OCDMA encoder in Fig. 2.2 for generating a time spread
1D OCDMA code, uses fiber delay lines, optical power splitters and optical power
combiners as shown in Fig. 2.8. A single input optical pulse from a laser is split
by using an optical splitter and each split pulse goes through different lengths of
optical fiber delay lines. The number of output ports of the optical splitter to
which the fiber delay lines are connected is determined by the weight (K) of the
1D OCDMA code sequence. Each delay line introduces a predetermined amount
of delay, varying from 0 to T, based on the chip positions of the OCDMA code.
The optical combiner at the end combines the split pulses with different delays to
generate the required 1D OCDMA code sequence.
Splitter
Power
Optical
Optical
Power
Combiner
8
2
0
Figure 2.8: An optical encoder to generate a time spread 1D OCDMA code
Figure 2.9 shows the generation of a 1D OCDMA code with weight K = 3
having pulses at chip locations 0, 2 and 8 using the OCDMA encoder shown in Fig.
2.8. Figure 2.9(a) represents an electrical pulse from the data to be transmitted,
Fig. 2.9(b) represents an optical pulse derived from the electrical pulse and Fig.
2.9(c) represents a time spread 1D OCDMA code. The optical pulse shown in
Fig. 2.9(b) passes through an optical power splitter to get split into three pulses,
24
2.2 Architecture of OCDMA Networks
which in turn enter different lengths of fiber delay lines. In Fig. 2.8 given above,
delays of 0, 2 and 8 time chips are shown. When the three pulses are combined in
the optical power combiner, the resultant output appears as shown in Fig. 2.9(c)
to represent an 1D OCDMA code.
5 10 15 0
t
(c)
(b)
(a)
Figure 2.9: Waveforms showing the generation of a 1D OCDMA code (c) from an
electrical pulse (a) converted into an optical pulse (b) from a laser
Figure 2.10 shows the structure of the OCDMA decoder used to despread the
code from the encoder (Fig. 2.8). The decoder for the same code would be similar
to the encoder except for the length of the fiber delay lines. These simple 1D
OCDMA encoders and decoders use 1 × K optical power splitters and K × 1
optical power combiners respectively.
Optical
Power
Combiner Splitter
Power
Optical
8
6
0
Figure 2.10: Optical decoding of the spread 1D OCDMA code shown in Fig. 2.8
Figure 2.11 shows how a 1D OCDMA code is decoded using the decoder shown
in Fig. 2.10. The 1D code is split by using an optical power splitter into three
codes. The three codes go through different lengths of fiber delay lines to get a
delay of 0, 6 and 8 time chips. Finally, when the three codes are added in the
25
OCDMA System Model
optical power combiner, the resultant output autocorrelation function has a peak
as shown in Fig. 2.11(d).
(c)
(b)
(a)
5 10 15 0
t
(d)
Figure 2.11: Waveforms showing the autocorrelation function (d) from addition of
the 1D OCDMA code with different delays (a, b, c) by using the OCDMA decoder
A 2D wavelength - time SPR OCDMA code can be generated by replacing the
blocks ‘Optical Power Splitter’ and ‘Optical Power Combiner’ in the 1D OCDMA
encoder (Fig. 2.8) with ‘Wavelength Demultiplexer’ and ‘Wavelength Multiplexer’
respectively. Decoding of the 2D wavelength - time SPR OCDMA code can be
achieved by modifying the 1D OCDMA decoder in the same way. An alternative
way of generating the 2D wavelength - time SPR OCDMA codes is by using a
series of fiber Bragg gratings (FBGs) with different lengths of fiber corresponding
to different time chips in between two FBGs. The reflections from the FBGs can
be collected by using an optical circulator giving a 2D SPR OCDMA code. The
structure of the decoder is similar to that of the encoder except for the position
of the FBGs reflecting each wavelength, which is in the reverse order as that of
the encoder.
A 2D wavelength - time MPR OCDMA code can be generated by using a 2D
OCDMA encoder shown in Fig. 2.12. The input to the encoder should be a multi-
wavelength optical pulse giving a 2D wavelength - time MPR OCDMA code at
the output. The 2D OCMDA encoder consists of 1D OCDMA encoders whose
number depends on the number of wavelengths used in the code. The spreading
in the wavelength domain is provided by the wavelength demultiplexer and the
26
2.2 Architecture of OCDMA Networks
spreading in the time domain is due to the 1D encoders. The decoding process
of these codes would be in the same manner as the encoding. The structure of
the 2D wavelength - time MPR decoder would be one in which the 1D OCDMA
encoders in Fig. 2.12 are replaced by 1D OCDMA decoders (Fig. 2.10).
1D OCDMA
Encoder
1D OCDMA
Encoder
1D OCDMA
Encoder
Demultiplexer
Wavelength
λ
1
λ
2
Multiplexer
Wavelength
λ
w
Figure 2.12: An optical encoder to generate a 2D MPR OCDMA code
Optical encoders for generating 3D space - wavelength - time SPP OCDMA
codes would have an optical power splitter at the output of a 2D wavelength -
time SPR encoder. Similarly, the generation of 3D space - wavelength - time
MPP OCDMA codes is possible by applying the output of 2D wavelength - time
MPR encoder to an optical power splitter. The number of output ports of the
optical power splitter is determined by the number of space channels used by the
3D code. The structure of the 3D decoder would have multiple 2D decoders, one
for each space channel. The outputs of all the 2D decoders are combined with an
optical power combiner to yield the autocorrelation peak. The use of optical hard
limiters at the output of each autocorrelator reduces the effect of multiple access
interference from other codes. Optical hard limiters are also referred to as optical
thresholders.
27
OCDMA System Model
2.3 Construction of OCDMA Code Families
New methods of constructing code families for OCDMA have been explored to
suit intensity modulation/direct detection (IM/DD) in optical fiber communica-
tions. Some reported methods of constructing 1D OCDMA code families, namely
prime sequences, optical orthogonal codes (OOCs) and balanced incomplete block
designs (BIBD) along with their properties are given in the following subsections.
The autocorrelation and crosscorrelation properties along with the advantages and
disadvantages of each code family are discussed. The probability of error due to
MAI [26] for some 1D OCDMA code families is elaborated. The construction and
properties of some reported 2D and 3D OCDMA code families are discussed in
Chapter 3.
2.3.1 Large Weight, Small Length Codes: Prime Sequences
References [11, 17] have reported that, for any prime number p, the length of the
code sequences would be T = p
2
, the weight would be K = p and the cardinality
would be N
max
= p. The maximum crosscorrelation between any two users is given
by λ
c
= 2. Reference [17] reports a maximum out-of-phase autocorrelation value
of 1 and termed it as the one coincidence property. A maximum crosscorrelation
of p in case of synchronous OCDMA is reported in [6]. Reference [25] reported 2
n
prime sequence codes, which are constructed by modifying the prime sequences
using a delay-distribution constraint. The weight of the codes is given by K = 2
n
and the properties for K = 4, 8, 16 and 32 are discussed. The autocorrelation and
crosscorrelation values of these codes are more than 1. Following is the method of
constructing prime sequences [17]:
1. Select a prime number p.
2. Write down the field elements in ascending or descending order.
28
2.3 Construction of OCDMA Code Families
3. Multiply this row by each field element modulo p to get p rows. Each row
denotes a sequence.
The sequences are of the form S
x
= (s
x0
, s
x1
, . . . , s
xj
, . . . , s
x(p−1)
). These sequences
are mapped [11] into a binary code sequence C
x
= (c
x0
, c
x1
, . . . , c
xi
, . . . , c
x(T−1)
),
by assigning ones in positions i = s
xj
+ jp for j = 0, 1, . . . , p − 1 and zeros in all
the other positions.
2.3.2 Minimum Correlated OOCs
One of the methods of verifying OOCs is the concept of extended sets [30]. The
formation of extended sets is based on a set of rules of addition of relative delays
between successive chips. By following the rules, it is possible to identify if two
codes have a crosscorrelation of one or more. Extended sets are formed by the
following rules: The relative delay sets of all codes are formed. Each delay set
comprises of delay elements, which are the delays between successive chips of a
code. The last delay element (difference between K-th and first chip position) is
obtained by modulo-T arithmetic. Extended set of a code is formed having the
sum of all connected delay elements along with the original delay set. The total
number of elements in an extended set is K(K −1). If two extended sets have no
element in common, then the two codes have a maximum crosscorrelation of ‘1’.
The formation of an extended set is also possible by building a difference table
(Table 2.1) of a cyclic difference set [109]. The difference table is built by the
modulo-T subtraction (⊖
T
) between elements of the difference set. The elements
of such a difference table are the elements of extended sets.
By extending the extended set concept, one can construct OOCs [27,32] having
maximum autocorrelation and crosscorrelation values of 1. Reference [27] gives a
method of constructing OOCs and examines the performance when the temporal
29
OCDMA System Model
Table 2.1: Difference table of a cyclic difference set (1,3,9) for T = 13 equivalent
to extended set

T
1 3 9
1 0 11 5
3 2 0 7
9 8 6 0
length of the codes is shortened. Shortening of the codes increases the crosscor-
relation value of the code family to 2. The method of constructing OOCs [32] is
described by the following algorithm.
1. Specify the weight, K, and number of users (= number of sequences), N
max
,
for the OOC. These are the only two input parameters (initially, the sequence
length, T, is unknown and will be determined at the end of the algorithm).
2. Assume that a pulse is placed at the zero-th chip position for all OOC
sequences.
3. The first delay elements of the OOC sequences are assigned as α
i,1
= i, for
i = 1, 2, ..., N
max
.
4. Start to form the extended sets, E
i
’s, by adding α
i,1
’s to these sets; i.e.,
initialize E
i
= {α
i,1
}, for i = 1, 2, ..., N
max
. Then, initialize i=1.
5. Initialize the offset parameter q = N
max
.
6. For the OOC sequence under consideration, S
i
, increment q, until its value
is not present in any of the partial extended sets.
7. Add q as a delay element for the OOC sequence under consideration, S
i
, iff
all linear combinations of jointly connected relative delay elements are not
repeated within the partial extended set under consideration or within the
partial extended sets of the other sequences. In case q is added as a delay
30
2.3 Construction of OCDMA Code Families
element for S
i
, then update the extended set for this sequence and go to step
9, if not, go to step 8.
8. Increment q and go to step 7.
9. If all K pulses for all N
max
sequences have been assigned (this is equivalent
to assigning all K1 delay elements for all N
max
sequences), then proceed to
step 10. If not, then move to the next OOC sequence (i.e., increment i; if
the incremented i is equal to N
max
+ 1, reset i = 1) and go to step 5.
10. The chip positions of the pulses for the N
max
OOC sequences are determined
using
S
i
= {0, α
i,1
, α
i,1

i,2
, α
i,1

i,2

i,3
, . . . , α
i,1

i,2
+. . . +α
i,(k−1)
}
for i = 1, 2, ..., N
max
.
11. Calculate the lower bound for the number of chips as “T
opt
= N
max
K(K −
1)+1”, which is the sequence length of perfect optimal OOCs. Starting from
T = T
opt
, increment T until the elements occur only once in the union of all
extended sets. The smallest T value satisfying these constraints is assigned
as the OOC sequence length.
2.3.3 Optimal Length, Minimum Correlated OOCs
The construction of 1D OOCs is also possible by using BIBD [110,111,112]. BIBD
is a very old tool of combinatorial theory, having a variety of applications in various
fields. The fundamental equations that govern a BIBD are:
b ×k = v ×r
31
OCDMA System Model
r ×(k −1) = λ ×(v −1)
where, b is the number of blocks in a design, k is the number of elements in a block
- equivalent to weight (K) of a code, v is the number of varieties - equivalent to the
temporal length T of the code, r is the number of times each variety is replicated
in the design, λ is the number of times each pair of elements occurs in the design
- equivalent to crosscorrelation (λ
c
) of codes in a code family.
The construction of BIBD code families consists of determining the initial
blocks, followed by modular addition to the initial blocks which gives all the
blocks (b). BIBD codes can be constructed in many ways. One of the methods of
constructing a family of N
max
initial blocks is as follows:
T = v = N
max
k(k −1) + 1
For triple systems ‘k = 3’, hence the above equation gives “T = 6N
max
+1”. x,
the primitive element of T is to be determined. The initial blocks are then given
by:
(x
0
, x
2Nmax
, x
4Nmax
), (x, x
2Nmax+1
, x
4Nmax+1
), . . . , (x
Nmax−1
, x
3Nmax−1
, x
5Nmax−1
)
Triple systems also exist for v = 6N
max
+3 and some others. BIBD codes with
k = 4 have v = 12N
max
+ 1 and v = 12N
max
+ 4. An advantage of BIBD is the
existence of codes for primes as well as non-primes. BIBD codes with k ≥ 4 are not
known to exist for some large values of v. For such cases, code construction can
be done using the extended set concept. An example of a Steiner Triple System
with length ‘13’ has initial blocks (1,3,9) and (2,5,6), which can be two codes in
an OCDMA code family.
32
2.3 Construction of OCDMA Code Families
For a given cardinality and weight, code families constructed using primes have
shorter temporal length and larger correlation compared to those constructed us-
ing the extended set concept or BIBD. Shorter temporal length is beneficial to
fabricate smaller devices for encoding and decoding, whereas larger correlation
leads to higher probability of error due to MAI. As indicated by the title of the
section, BIBD is advantageous over extended sets in terms of the optimum tem-
poral length.
2.3.4 Probability of Error Due to MAI
The probability of error due to MAI of 1D OCDMA code families having a max-
imum crosscorrelation value of 1 [26] is elaborately discussed with the help of
an example. These calculations are based on assumptions of a positive and ad-
ditive optical fiber communication system, which have been considered in [26].
The derivation considers chip synchronization between asynchronous users. This
consideration gives us an upper bound of probability of error due to MAI [31].
The probability of erroneous detection arises only when the desired user code is
not being transmitted, i.e., the user transmits a 0. The probability of the desired
user (U
j
) transmitting a 0 is given by
P(U
j
: 0) =
1
2
. (2.1)
For N interfering users, the combination of users overlapping with K pulses
out of K pulses of the desired user which may or may not cause an error due to
the presence of the interfering users’ pulses at different chip locations or at the
same chip location respectively is given by
U
j
[error/no error] =
N

j=K
N
C
j
. (2.2)
33
OCDMA System Model
The combination of users overlapping with a combination of K pulses of the
desired user which would cause an error due to the presence of the interfering
users’ pulses at different chip locations is given by
U
j
[error] =
N

j=K
_
N
C
j
_ _
j
C
K
_
. (2.3)
The argument leading to Eqn. (2.3) is explained with the help of an example
in Appendix. The total combinations of N interfering users is 2
N
. The probability
that N interfering users overlap K pulses of the desired user to cause an error is
P(U
j
[error]) =
1
2
N
N

j=K
_
N
C
j
_ _
j
C
K
_
. (2.4)
The probability of all K pulses of the desired user being overlapped by K
pulses of different interfering users is
P(K|K) =
_
K
T
_
K
K!. (2.5)
Combining (2.1), (2.4) and (2.5), the upper bound of 1D probability of error
due to MAI is given by
P
e
=
1
2
N+1
N

j=K
_
N
C
j
_ _
j
C
K
_
_
K
T
_
K
K!. (2.6)
Equation (2.6) is a simplified reproduction of the equation for hard-limiting re-
ceiver, assuming chip synchronous case for a threshold equal to the weight [26].
Equation (2.6) has also been verified by simulating the crosscorrelation of 1D
OOCs having weight K = 3 for ‘4’ interfering codes with T = 31, ‘4 and 5’ inter-
fering codes with T = 43 and for K = 2, T = 15 with ‘2, 3, 4, 5 and 6’ interfering
codes. Extension of this result is used for probability of error due to MAI calcu-
lations in Sections 3.5.2 and 3.6.2 of Chapter 3 and in Sections 4.4.2 and 4.5.2 of
34
2.3 Construction of OCDMA Code Families
Chapter 4.
Following is the demonstration of eqn. (2.6) with the help of an example. Con-
sider a weight K = 2 OOC family with four users (U
0
, U
1
, U
2
, U
3
). The temporal
length of the OOC family is T = 9. The time-spread sequences of the four users
is shown in Table 2.2. The time chips of the sequences are indicated by t
1
to t
9
.
Table 2.2: Example: four user, K = 2 OOC family
t
1
t
2
t
3
t
4
t
5
t
6
t
7
t
8
t
9
U
0
1 0 0 0 0 0 0 0 1
U
1
0 1 0 0 0 0 0 1 0
U
2
0 0 1 0 0 0 1 0 0
U
3
0 0 0 1 0 1 0 0 0
Considering U
0
as the desired user, whose probability of error due to MAI is to
be determined, the interfering users would be U
1
, U
2
, U
3
. The number of interfering
users in this case is N = 3. The combination of users overlapping with K pulses
of U
0
which may or may not cause an error are
U
1
U
2
, U
1
U
3
, U
2
U
3
, U
1
U
2
U
3

3
C
2
+
3
C
3
=
N

j=K
N
C
j
. (2.7)
The user combination U
1
U
2
would not cause an error if U
1
and U
2
overlap
at the same chip position. The user combination U
1
U
2
would cause an error if
U
1
overlaps at chip t
1
and U
2
overlaps at chip t
9
or U
1
overlaps at chip t
9
and
U
2
overlaps at chip t
1
. This is represented as U
(a)
1
U
(b)
2
, where
(a)
represents t
1
|t
9
and
(b)
represents t
9
|t
1
. The combination of users and their combination of pulses
overlapping with each of the K pulses of U
0
to cause an error are
35
OCDMA System Model
U
(a)
1
U
(b)
2
, U
(a)
1
U
(b)
3
, U
(a)
2
U
(b)
3
, (U
1
U
2
)
(a)
U
(b)
3
, (U
1
U
3
)
(a)
U
(b)
2
,
U
(a)
1
(U
2
U
3
)
(b)

_
3
C
2
_ _
2
C
2
_
+
_
3
C
3
_ _
3
C
2
_
=
N

j=K
_
N
C
j
_ _
j
C
K
_
. (2.8)
2.4 Feasibility of Getting Large Delay in OCDMA
Code Generation
The generation of 1D OCDMA codes involves the use of optical fiber delay lines as
discussed in Sec. 2.2. The amount of delay generated by an optical fiber is ln/c,
where l is the length of the optical fiber, n is the refractive index of the optical
fiber core and c is the velocity of light. Since c is constant and n is usually the
same for optical fibers, l is changed to get the required amount of delay. For a
Gbps OCDMA network with large cardinality, the delay necessary is of the order
of a few picoseconds or in the sub-picosecond region. For such delay between
pulses, the required differential length of optical fibers is in the sub-micron range.
Sub-micron precision lengths of optical fibers are difficult to cut in practice. The
use of optical fibers as delay elements are more practical in the nanosecond region,
which would need differential lengths of the order of centimeters.
For a high-speed OCDMA network, it is more practical to use 2D or 3D systems
with optical fiber delay lines. Sub-micron differential lengths can be fabricated on
planar lightwave circuits or optical integrated circuits. Various integrated-optic
technologies to realize optical integrated circuits are based on substrates of gallium
aluminum arsenide, gallium indium arsenide phosphide, silica on silicon, silicon,
polydimethylsiloxane (PDMS), polymethylmethacrylate (PMMA), SU-8, lithium
36
2.5 Applications of Lithium Niobate Devices
niobate and others. Higher the refractive index of the substrate, larger is the de-
lay. Gallium aluminum arsenide, gallium indium arsenide phosphide and silicon
substrates have refractive indices greater than three in the optical communication
window. The refractive index of silica is of the order of 1.5 in the optical com-
munication window. The refractive indices of PDMS, PMMA and SU-8 are lower
than that of silica. Lithium niobate substrate has a refractive index of the order
of 2.2 in the optical communication window.
Substrates of gallium aluminum arsenide, gallium indium arsenide phosphide
and silicon would yield smaller devices compared to lithium niobate, silica, PDMS,
PMMA and SU-8 due to the shorter differential lengths required to get a given
amount of delay between the optical pulses. The achievable delay between the
optical pulses on lithium niobate is explored in this thesis.
2.5 Applications of Lithium Niobate Devices
Lithium niobate (LiNbO
3
) is a compound of niobium, lithium and oxygen, is a
colorless solid and is insoluble in water. It is transparent for wavelengths between
350 and 5200 nanometers and has negative uniaxial birefringence. It’s ordinary
and extra-ordinary refractive indices depend on the wavelength of operation and
are governed by the Sellmeier equations. The ordinary refractive index varies
from ≈ 2.4 at a wavelength of 420 nanometers to ≈ 2.1 at a wavelength of 4000
nanometers. The extra-ordinary refractive index varies from≈ 2.3 at a wavelength
of 420 nanometers to ≈ 2.0 at a wavelength of 4000 nanometers.
It can be doped with various materials depending on the application. In optical
communication applications, lithium niobate is usually doped with titanium, nickel
and erbium among others. Among the many dopants, titanium indiffused lithium
niobate waveguides have the lowest transmission loss. Erbium doping is used for
37
OCDMA System Model
amplification applications in the 1550 nanometers optical communication window.
The ferroelectric property of lithium niobate makes it an ideal choice for switch-
ing applications [113, 114, 115]. Further advances in titanium indiffused lithium
niobate devices include Y-junction splitters, Y-junction combiners, directional
couplers and two-mode interference couplers/zero-gap directional couplers among
others. Titanium indiffused lithium niobate directional coupler based WDM mul-
tiplexers and demultiplexers are also reported. In this thesis, we explore the
application of titanium indiffused lithium niobate zero-gap directional couplers
and Y-junction 3dB power splitters in generating optical spreading codes.
Beam propagation method (BPM) [116, 117] is the commonly used tool to
simulate lithium niobate based devices. Effective index based matrix method
(EIMM) [118,119,120] is also used to simulate titanium indiffused lithium niobate
devices. We have used EIMM to simulate the functionality of the devices proposed
in Chapter 5. The use of lithium niobate as an optical delay element in the
generation of optical spreading codes is explored, owing to its higher refractive
index in comparison with optical fibers.
2.6 Statement of the Problem
Most of the OCDMA code families reported in literature have a high probability of
error due to MAI at full cardinality. The high probability of error can be reduced
by increasing the weight of the code families. An increase in the code weight,
drastically increases the code dimension of OCDMA code families. Increased code
dimension leads to complex hardware to generate the codes. The hardware re-
quirement in generating the codes with large weight and large code dimension is
in the form of optical splitters or optical wavelength demultiplexers/filters with
more output ports corresponding to the weight of the code, increased length of
38
2.6 Statement of the Problem
optical fiber delay lines depending on the temporal length of the code families
and optical combiners or optical wavelength multiplexers with more input ports
to get the optically encoded signal. Hence, different constructions of OCDMA
code families with low probability of error due to MAI at full cardinality for low
code weight and low code dimension is proposed in Chapters 3 and 4.
Moreover, most of the code generators, as given in Section 2.2.2, use optical
fiber components like optical fiber delay lines and optical fiber Bragg gratings.
The use of such optical fiber components makes the OCDMA system bulky. The
bulky components limit the technology from being deployed on a large scale. The
generation of proposed OCDMA code families having low weight and low code
dimension is explored with titanium indiffused lithium niobate integrated optic
technology in Chapter 5.
39
C H A P T E R 3
OCDMA Code Families based on
a Novel RWOP Algorithm
3.1 Introduction
Various constructions of OCDMA code families are reported in literature which
can be broadly classified based on the type of spreading used. Among them, most
have concentrated on the performance of the code families based on their spectral
efficiency and probability of error due to multiple access interference (MAI). Two-
dimensional (2D) code families spread in the wavelength - time domain [57, 64, 68,
121] have lower probability of error due to MAI compared to one-dimensional (1D)
code families. Similarly, three-dimensional (3D) code families [102, 105, 106, 107]
have lower probability of error due to MAI compared to 2D code families. In
this chapter, the construction and analysis of some of the existing 2D and 3D
OCDMA code families is discussed which is followed by the proposal of a row-wise
orthogonal pairs (RWOP) algorithm for wavelength and/or spatial allocation. The
OCDMA Code Families based on a Novel RWOP Algorithm
RWOP algorithm is applied to construct 2D multipulse per row (MPR) and 3D
multipulse per plane (MPP) code families. The investigated performance metrics
of the proposed RWOP-based code families are cardinality, spectral efficiency for
different code dimensions and the probability of error due to MAI for different
numbers of active users. The maximum number of users supported by a code
family is termed here as its cardinality. The ratio between cardinality to the
code dimension of a code family is termed as its spectral efficiency. The code
dimension of a 1D OCDMA code family spread in the time domain would be the
number of time chips. For 2D wavelength-time code families, the code dimension
is the product of number of wavelengths and the number of time chips, and for
3D space-wavelength-time code families, the code dimension is the product of
number of spatial channels, number of wavelengths and the number of time chips.
The above performance metrics of the constructed code families are analyzed and
compared with existing 2D [57,63,65,66] and 3D [102] code families. A comparison
between the constructed 2D and 3D code families is also shown. All comparisons
are based on equivalent code dimensions.
The construction and performance analysis of the existing 2D and 3D code
families are shown in Sec. 3.2 and 3.3 respectively. Section 3.4 discusses the
RWOP algorithm for wavelength and/or spatial allocation. Sections 3.5 and 3.6
show the application of RWOP algorithm in constructing 2D and 3D code families
and their performance analysis followed by summary in Sec. 3.7.
3.2 Two-Dimensional OCDMA Code Families
Among the several methods of constructing 2D code families, the code families
with which the proposed code families have been compared are shown. The fol-
lowing subsections contain the construction and properties of generalized multi-
42
3.2 Two-Dimensional OCDMA Code Families
wavelength prime codes (GMWPCs) [57], generalized multi-wavelength Reed-
Solomon codes (GMWRSCs) [57], multi-wavelength OOCs (MWOOCs) [65] and
folded OOCs (FOOCs) [66].
3.2.1 GMWPCs
The GMWPC families are generated with the help of a set of prime numbers.
These code families can be constructed for any integer number of wavelengths (W)
and the number of time chips (T) is a product of primes. A set of prime numbers
p
k
≥ p
k−1
≥ . . . ≥ p
1
≥ W gives the number of time chips (T = p
1
p
2
. . . p
k
). The
code C
k
, consisting of the 2D blocks
(0, 0), (1, i
1
+i
2
p
1
+· · · +i
k
p
1
p
2
. . . p
k−1
),
(2, 2 ⊙
p
1
i
1
+. . . + (2 ⊙
p
k
i
k
)p
1
p
2
. . . p
k−1
), . . . ,
(W −1, (W −1) ⊙
p
1
i
1
+ ((W −1) ⊙
p
2
i
2
)p
1
+. . .+ ((W −1) ⊙
p
k
i
k
)p
1
p
2
. . .p
k−1
)
is an (W × T, K

, 0, 1) GMWPC with N
max
(GMWPC) = T codewords of size
W × T, zero autocorrelation side lobes, crosscorrelation value of at most 1 and
weight K

= W, where “⊙
p
k
” represents modulo-p
k
multiplication and i
1
=
{0, 1, 2, . . . , p
1
− 1}, i
2
= {0, 1, 2, . . . , p
2
− 1}, . . . , i
k
= {0, 1, 2, . . . , p
k
− 1}. Each
pair in parenthesis represents the position of an optical pulse chip in a 2D wave-
length - time array. The first number is the wavelength index and the second
number is the time chip index. It should be noted that, the wavelength indices
in each block are successive integers and the time chip indices are derived using
primes. The spectral efficiency of the GMWPCs is 1/W, which is the inverse of
the weight of the GMWPCs.
An example GMWPC construction for W = 4, T = 7 and K

= 4 is shown in
43
OCDMA Code Families based on a Novel RWOP Algorithm
Table 3.1. The wavelengths are represented as w
1
, w
2
, w
3
and w
4
while the time
chips are shown as t
1
, t
2
, . . . , t
7
.
Table 3.1: Example 2D GMWPC construction for W = 4, T = 7 & K

= 4
Users Wavelength and time chip allocation (W : T)
U
1
(w
1
: t
1
), (w
2
: t
1
), (w
3
: t
1
), (w
4
: t
1
)
U
2
(w
1
: t
1
), (w
2
: t
2
), (w
3
: t
3
), (w
4
: t
4
)
U
3
(w
1
: t
1
), (w
2
: t
3
), (w
3
: t
5
), (w
4
: t
7
)
U
4
(w
1
: t
1
), (w
2
: t
4
), (w
3
: t
7
), (w
4
: t
3
)
U
5
(w
1
: t
1
), (w
2
: t
5
), (w
3
: t
2
), (w
4
: t
6
)
U
6
(w
1
: t
1
), (w
2
: t
6
), (w
3
: t
4
), (w
4
: t
2
)
U
7
(w
1
: t
1
), (w
2
: t
7
), (w
3
: t
6
), (w
4
: t
5
)
3.2.2 GMWRSCs
These 2D code families are generated using Reed-Solomon (RS) codes. An RS
code of length T
rs
= p − 1 over GF(p) (Galois field of a prime number p) is a
cyclic code with a generator polynomial g(x) = (x−˜ α
b
)(x−˜ α
b+1
) . . . (x−˜ α
b+δ−2
),
in which ˜ α is a primitive element of GF(p), δ is the minimum distance, and the
integer b determines the starting terms of g(x). A [T
rs
, M, δ] RS code can be
represented in polynomial form as c(x) = c
0
+ c
1
x + . . . + c
j
x
j
+ . . . + c
p−2
x
p−2
,
where c
j
∈ GF(p) is the polynomial coefficient and M = T
rs
− δ + 1 is the code
dimension. The code cardinality N
max−rs
is usually related to the dimension M
as N
max−rs
= p
M
.
When δ = T
rs
−1 (i.e., M = 2) and b = 1, g(x) = (x−˜ α)(x−˜ α
2
) . . . (x−˜ α
p−3
).
For an RS codeword denoted as a
0
= (a
0
, a
1
, . . . , a
j
, . . . , a
p−2
), where a
j
∈ GF(p),
p codewords are obtained. The p codewords are a
i
= (a
i,0
, a
i,1
, . . . , a
i,j
, . . . , a
i,p−2
)
with i = {0, 1, . . . , p−1}, where a
i,j
= a
j
⊕i and “⊕” represents a modulo-p addi-
tion. All these codewords are candidates for the construction of the GMWRSCs.
44
3.2 Two-Dimensional OCDMA Code Families
For a set of prime numbers p
k
≥ p
k−1
≥ . . . ≥ p
2
≥ W, the 1D RS codewords
a
i
for i = [0, p −1] give 2D blocks
(a
i,0
, 0), (a
i,1
, 1), . . . , (a
i,j
, j), . . . , (a
i,W−2
, W −2)
which are the bases of the generalized code.
Then, the code C
k
, consisting of the blocks
(a
i,0
⊕l, ((a
i,0
⊕l) ⊙
p
2
i
2
)(W −1) +. . .+ ((a
i,0
⊕l) ⊙
p
k
i
k
)(W −1)p
2
. . .p
k−1
),
(a
i,1
⊕l, 1 + ((a
i,1
⊕l) ⊙
p
2
i
2
)(W −1) +. . .+ ((a
i,1
⊕l) ⊙
p
k
i
k
)(W −1)p
2
. . .p
k−1
),
(a
i,p−2
⊕l, (p −2) + ((a
i,p−2
⊕l) ⊙
p
2
i
2
)(W −1) +. . .
+((a
i,p−2
⊕l) ⊙
p
k
i
k
)(W −1)p
2
. . . p
k−1
)
is an (W × T, K

, 0, 1) GMWRSC with temporal length T = (W − 1)p
2
. . . p
k
,
N
max
(GMWRSC) = WT/K

codewords of size W × T, zero autocorrelation
side lobes, crosscorrelation value of at most 1 and weight K

= W − 1, where
l = {0, 1, 2, . . . , W−1}, i
2
= {0, 1, 2, . . . , p
2
−1}, i
3
= {0, 1, 2, . . . , p
3
−1}, . . . , i
k
=
{0, 1, 2, . . . , p
k
− 1}. In this case, both the wavelength indices and the time chip
indices in each block are derived using the RS codes. The spectral efficiency of the
GMWRSCs is 1/(W −1), which is the inverse of the weight of the GMWRSCs.
An example GMWRSC construction for W = 3, T = 10 and K

= 2 is shown
in Table 3.2. The wavelengths are represented as w
1
, w
2
and w
3
while the time
chips are shown as t
1
, t
2
, . . . , t
10
.
The probability of error due to MAI of (W ×(W −1)W, K

, 0, 1) GMWRSCs
is given by
P

e
=
1
2
N

i=K

N
C
i
_
K
′2
2WT
_
i
_
1 −
K
′2
2WT
_
N−i
. (3.1)
45
OCDMA Code Families based on a Novel RWOP Algorithm
Table 3.2: Example 2D GMWRSC construction for W = 3, T = 10 & K

= 2
Users W : T Users W : T Users W : T
U
1
(w
3
: t
1
), (w
2
: t
2
) U
6
(w
1
: t
1
), (w
3
: t
2
) U
11
(w
2
: t
1
), (w
1
: t
2
)
U
2
(w
3
: t
5
), (w
2
: t
4
) U
7
(w
1
: t
1
), (w
3
: t
6
) U
12
(w
2
: t
3
), (w
1
: t
2
)
U
3
(w
3
: t
9
), (w
2
: t
6
) U
8
(w
1
: t
1
), (w
3
: t
10
) U
13
(w
2
: t
5
), (w
1
: t
2
)
U
4
(w
3
: t
3
), (w
2
: t
8
) U
9
(w
1
: t
1
), (w
3
: t
4
) U
14
(w
2
: t
7
), (w
1
: t
2
)
U
5
(w
3
: t
6
), (w
2
: t
10
) U
10
(w
1
: t
1
), (w
3
: t
8
) U
15
(w
2
: t
9
), (w
1
: t
2
)
3.2.3 MWOOCs
The MWOOC families are constructed using 1D OOCs and prime sequences. The
MWOOC families contain a combination of SPR and MPR codes. The weight of
the MWOOCs is the same as that of the 1D OOCs used in the construction (K

=
K). Wavelength-time coding schemes, using multiple wavelengths to represent
pulses in time slots, can be represented as W × T matrices, where W is the
number of rows (or available wavelengths), and T is the number of columns (or
time chips) [65]. A time chip in a matrix of the MWOOCs contains either nothing
or one pulse (of one wavelength). The wavelength used in a time chip is determined
by the permutation of wavelengths, algebraically controlled by prime sequences
over Galois field, onto the nonzero time chips of the codewords of a time-spreading
optical code. The scheme also works with non prime integers, as long as the code
weight is no greater than the smallest prime factor of the non prime integer. To
keep MAI as low as possible, optical codes with crosscorrelation value of at most
1, such as the OOCs are used as the time-spreading codes.
Each sequence (or row) is used as a seed for a group of new matrices. The code
weight K of the OOC can be as large as p, giving a total N
group
= p groups of
prime sequences, from which new matrices can be formed. By mapping these
prime sequences with W = p wavelengths and the number of time-spreading
codewords N
OOC
(of weight K ≤ W), there are totally N
max
(OOC)N
group
W =
46
3.2 Two-Dimensional OCDMA Code Families
N
max
(OOC)p
2
matrices. As a general rule, for K ≤ p, the first K elements in
each prime sequence are used in the permutations.
For a positive integer p

= p
1
p
2
. . . p
k
, where p
1
≤ p
2
≤ . . . ≤ p
k
are prime
numbers for a positive integer k. p

prime sequences can be constructed from
GF(p

) with non repeated elements if the code weight of the OOC is less than
or equal to p
1
(i.e., K ≤ p
1
), giving N
group
= p

groups of prime sequence, from
which new matrices can be formed. By mapping these prime sequences with p

wavelengths and the number of time-spreading codewords N
max
(OOC), there are
in total N
max
(OOC)N
group
p

= N
max
(OOC)p
′2
matrices. This construction has
out of phase autocorrelation value of 1 and crosscorrelation value of at most 1.
The cardinality of the MWOOCs (N
max
(MWOOC)) is
1. If the number of available wavelengths is a prime p and code weight K ≤ p,
then N
max
(MWOOC) = N
max
(OOC)p
2
, which gives the largest cardinality.
2. If the number of available wavelengths is a positive integer p

= p
1
p
2
. . . p
k
and code weight K ≤ p
1
, then N
group
= p

and N
max
(MWOOC) = N
max
(OOC)p
′2
,
which gives the largest cardinality.
3. If the number of available wavelengths is a positive integer p

= p
1
p
2
. . . p
k
and code weight K > p
1
, then we should use the largest prime number p
less than or equal to p

, such that p ≥ K > p
1
and N
max
(MWOOC) =
N
max
(OOC)p
2
.
An example MWOOC construction for W = 3, T = 9 and K

= 2 is shown in
Table 3.3. This construction uses four 1D OOCs having K = 2, T = 9, and prime
sequences from GF(9). The wavelengths are represented as w
1
, w
2
and w
3
while
the time chips are shown as t
1
, t
2
, . . . , t
9
.
47
OCDMA Code Families based on a Novel RWOP Algorithm
Table 3.3: Example 2D MWOOC construction for W = 3, T = 9 & K

= 2
Users W : T Users W : T Users W : T
U
1
(w
1
: t
1
t
9
) U
13
(w
1
: t
1
), (w
2
: t
9
) U
25
(w
1
: t
1
), (w
3
: t
9
)
U
2
(w
2
: t
1
t
9
) U
14
(w
2
: t
1
), (w
3
: t
9
) U
26
(w
2
: t
1
), (w
1
: t
9
)
U
3
(w
3
: t
1
t
9
) U
15
(w
3
: t
1
), (w
1
: t
9
) U
27
(w
3
: t
1
), (w
2
: t
9
)
U
4
(w
1
: t
2
t
8
) U
16
(w
1
: t
2
), (w
2
: t
8
) U
28
(w
1
: t
1
), (w
3
: t
8
)
U
5
(w
2
: t
2
t
8
) U
17
(w
2
: t
2
), (w
3
: t
8
) U
29
(w
2
: t
1
), (w
1
: t
8
)
U
6
(w
3
: t
2
t
8
) U
18
(w
3
: t
2
), (w
1
: t
8
) U
30
(w
3
: t
1
), (w
2
: t
8
)
U
7
(w
1
: t
3
t
7
) U
19
(w
1
: t
3
), (w
2
: t
7
) U
31
(w
1
: t
1
), (w
3
: t
7
)
U
8
(w
2
: t
3
t
7
) U
20
(w
2
: t
3
), (w
3
: t
7
) U
32
(w
2
: t
1
), (w
1
: t
7
)
U
9
(w
3
: t
3
t
7
) U
21
(w
3
: t
3
), (w
1
: t
7
) U
33
(w
3
: t
1
), (w
2
: t
7
)
U
10
(w
1
: t
4
t
6
) U
22
(w
1
: t
4
), (w
2
: t
6
) U
34
(w
1
: t
1
), (w
3
: t
6
)
U
11
(w
2
: t
4
t
6
) U
23
(w
2
: t
4
), (w
3
: t
6
) U
35
(w
2
: t
1
), (w
1
: t
6
)
U
12
(w
3
: t
4
t
6
) U
24
(w
3
: t
4
), (w
1
: t
6
) U
36
(w
3
: t
1
), (w
2
: t
6
)
The probability of error due to MAI of the MWOOCs is given by
P

e
=
1
2
K


i=0
(−1)
i
_
K

C
i
_
_
1 −
qi
K

_
N−1
, (3.2)
where q =
1
p
K
′2
(Nmax(OOC)p−1)
2Tooc(Nmax(OOC)p
2
−1)
+
p−1
p
K
′2
(Nmax(OOC)p−1)+(K

−1)
2
2Tooc(Nmax(OOC)p
2
−1)
, N is the number
of active users, N
max
(OOC) is the cardinality and T
ooc
is the temporal length of
the OOC used.
3.2.4 FOOCs
The construction of FOOC families is by folding of 1D OOCs. The FOOC fam-
ilies also contain a combination of SPR and MPR codes. The construction of
(W × T, K

, 1, 1) FOOCs [66] uses a (T
ooc
, K, 1, 1) 1D OOC family, where W is
the number of wavelengths and T is the code length of the FOOCs, K = K

48
3.2 Two-Dimensional OCDMA Code Families
is the weight of both OOCs and FOOCs and T
ooc
must be equal to W × T. If
N
max
(OOC) ≤
WT−1
K(K−1)
1D OOCs are used, the cardinality of the FOOCs is given
by N
max
(FOOC) = WN
max
(OOC). For an OOC family C
n
, the FOOC family is
obtained as A
n,s
(i, j) = C
n
(i ⊕
T
jW⊕
T
s), where n = 0, 1, . . . , N
max
(OOC) − 1,
0 ≤ s ≤ W −1, 0 ≤ i ≤ W −1 and 0 ≤ j ≤ T −1.
An example FOOC construction for W = 3, T = 9 and K

= 4 is shown in
Table 3.4. This construction uses two 1D OOCs having K = 4 and T = 27. The
wavelengths are represented as w
1
, w
2
and w
3
while the time chips are shown as
t
1
, t
2
, . . . , t
9
.
Table 3.4: Example 2D FOOC construction for W = 3, T = 9 & K

= 4
Users W : T
U
1
(w
1
: t
1
), (w
2
: t
1
t
2
t
4
)
U
2
(w
1
: t
1
t
2
t
4
), (w
3
: t
9
)
U
3
(w
2
: t
9
), (w
3
: t
1
t
3
t
9
)
U
4
(w
1
: t
1
t
6
), (w
2
: t
3
), (w
3
: t
1
)
U
5
(w
1
: t
3
), (w
2
: t
1
), (w
3
: t
5
t
9
)
U
6
(w
1
: t
1
), (w
2
: t
5
t
9
), (w
3
: t
2
)
The probability of error due to MAI of the FOOCs is given by
P

e
=
1
2
N−1

u=K

_
N−1
C
u
_
P
u
hit
(1 −P
hit
)
N−1−u
, (3.3)
where N is the number of active users and P
hit
=
_
(K
′2
−K)(WT−T)
WT−1
+K
′2
(Nmax(OOC)−1)
2T(Nmax(FOOC)−1)
_
.
A comparison of the probability of error due to MAI of the FOOCs with MWOOCs
shows similar performance.
49
OCDMA Code Families based on a Novel RWOP Algorithm
3.3 Three-Dimensional OCDMA Code Families
Space-wavelength-time 3D codes [102] constructed by extending 2D GMWPCs -
wavelength-time codes or space-time codes are discussed. Assuming (S×T, S, 0, 1)
space-time 2D codes, another degree of freedom can be added if multi-wavelength
light sources (W) are available for each spatial channel. The wavelength of each
spatial channel has to be chosen in code construction. For a given temporal
distribution of pulses over spatial channels, i.e., one codeword of the space/time
2D code, many different codewords can be generated by changing the wavelength
of pulse in each spatial channel. In assigning wavelength to each spatial channel,
orthogonality has to be maintained by constraining crosscorrelation between any
two codewords with different temporal distributions of pulses over spatial channels
to at most 1. The crosscorrelation between any two codewords with the same
temporal distribution is not greater than 1 because of the orthogonality of the 2D
code.
To extend 2D codes to 3D codes without losing orthogonality, wavelengths are
assigned to each spatial channel in such a way that any two distinct codewords
have no more than one spatial channel of the same wavelength. The same algo-
rithm that is used to assign temporal locations of pulses over spatial channels in
construction of the 2D GMWPCs is employed. Space-wavelength-time 3D code
can be constructed by applying 2D construction algorithm separately to space-
wavelength plane and space-time plane.
Distinct codewords (W) for each temporal distribution of pulses over spatial
channels are generated by applying the 2D GMWPC algorithm for assigning W
wavelengths over spatial channels. If the number of wavelengths is greater than
or equal to that of spatial channels (W ≥ S), due to the orthogonality of the
2D GMWPC construction algorithm, every cyclic shift in wavelength domain also
generates another codeword that is orthogonal to others. Therefore, W
2
T code-
50
3.4 The RWOP Algorithm
words can be generated by extending (S ×T, S, 0, 1) code to (S ×W ×T, S, 0, 1)
code if W ≥ S and the spectral efficiency is W/S. If W < S, cyclic shift of
codewords in wavelength domain coincide with other codewords, and thus, WT
codewords are possible for (S ×W ×T, S, 0, 1) 3D code and the spectral efficiency
is 1/S, which is the inverse of the weight of the SPP code families.
The probability of error due to MAI of the 3D SPP code families is given by
P
′′
e
=
1
2
N−1

i=S
_
N−1
C
i
_
_
S
2TW
_
i
_
1 −
S
2TW
_
N−1−i
. (3.4)
Space-wavelength-time 3D MPP code families are also constructed by using
the same 2D GMWPC construction algorithm. The 3D (S × W × T, SW, 0, 1)
code families have SW space - wavelength channels and SW pulses are allocated
such that there is a coincidence of optical pulses only in one space - wavelength
plane for any two distinct codewords.
3.4 The RWOP Algorithm
The RWOP algorithm proposed in this thesis is explained with the help of a
flowchart shown in Fig. 3.1 along with an example. The number of wavelengths
(W) to be used in the system is the first parameter to be chosen, an even number
greater than ‘2’. By choosing ‘2’, we get a single pair representing a single user
(not useful, considering the application of the code family to a multiple access
network). We demonstrate the algorithm for W = 6. All possible pairs
W
C
2
,
represented as (a, b) in Fig. 3.1 are generated. The array elements shown in the
below example represent wavelength indices.
The symbol ⊖
W
in Fig. 3.1 denotes modulo - W subtraction. The array [P
i
]
has dimension W×2, except for i = W/2 when [P
i
] has dimension
W
2
×2. The array
51
OCDMA Code Families based on a Novel RWOP Algorithm
START
W=Even no., i=1
Generate
W
C
2
pairs (a, b)
Collect pairs [P
i
] with a ⊖
W
b = i
i < W/2 i = i + 1
i=1
[P
i1
] ← UEP of [P
i
] [P
it
] ← REP of [P
i
]
[P
i2
] ← UEP of [P
it
]
[P
ia
] ← Put pairs in a row [P
ib
] ← Put pairs in a row
[A
2i−1
; A
2i
] ←[P
ia
; P
ib
]
i = i + 1 i < W/2
STOP
No [P]
Yes
Yes
No [A]
Figure 3.1: Flowchart depicting the proposed RWOP algorithm; UEP represents
unrepeated element-pairs, REP represents repeated element-pairs
[P], is formed by vertical concatenation of [P
i
]’s and has dimension
W(W−1)
2
× 2.
For W = 6, [P
i
]’s are represented as follows:
52
3.4 The RWOP Algorithm
P
1
=
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
1 2
2 3
3 4
4 5
5 6
6 1
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
P
2
=
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
1 3
2 4
3 5
4 6
5 1
6 2
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
P
3
=
_
¸
¸
¸
_
1 4
2 5
3 6
_
¸
¸
¸
_
.
The resultant array [P], is formed by vertical concatenation of [P
i
]’s and has
dimension
W(W−1)
2
×2.
An unrepeated element-pair (UEP) denotes either element of a pair is not
present in the accumulating target array ([P
i1
] or [P
i2
] shown in Fig. 3.1). A re-
peated element-pair (REP) denotes either or both elements of a pair are present in
the target array. Unrepeated element-pairs of [P
i
] are accumulated in [P
i1
] and the
repeated element-pairs are accumulated in the temporary array [P
it
]. Unrepeated
element-pairs of [P
it
] are then accumulated in [P
i2
] and repeated element-pairs are
discarded. [P
i1
] and [P
i2
] are converted into single-row vectors and stored in [P
ia
]
and [P
ib
] respectively. [P
ia
] and [P
ib
] are accumulated in successive rows of the
array [A
i
] respectively. In the example, [P
i1
], [P
it
], [P
i2
], [P
ia
], [P
ib
] and [A
i
] have
elements as shown below.
P
11
=
_
¸
¸
¸
_
1 2
3 4
5 6
_
¸
¸
¸
_
P
1t
=
_
¸
¸
¸
_
2 3
4 5
6 1
_
¸
¸
¸
_
P
12
=
_
¸
¸
¸
_
2 3
4 5
6 1
_
¸
¸
¸
_
P
1a
=
_
1 2 3 4 5 6
_
, P
1b
=
_
2 3 4 5 6 1
_
, A
1
=
_
_
1 2 3 4 5 6
2 3 4 5 6 1
_
_
53
OCDMA Code Families based on a Novel RWOP Algorithm
P
21
=
_
_
1 3
2 4
_
_
P
2t
=
_
¸
¸
¸
¸
¸
¸
_
3 5
4 6
5 1
6 2
_
¸
¸
¸
¸
¸
¸
_
P
22
=
_
_
3 5
4 6
_
_
P
2a
=
_
1 3 2 4 0 0
_
, P
2b
=
_
3 5 4 6 0 0
_
, A
2
=
_
_
1 3 2 4 0 0
3 5 4 6 0 0
_
_
P
31
=
_
¸
¸
¸
_
1 4
2 5
3 6
_
¸
¸
¸
_
P
3a
=
_
1 4 2 5 3 6
_
A
3
=
_
1 4 2 5 3 6
_
.
For a single value of i, pairs are arranged in two rows, except for i = W/2
when the pairs are arranged in a single row. Sorting is repeated twice for each i,
except for i = W/2. Some of the pairs are discarded during the sorting process to
make sure that no number is repeated in each row, i.e., the wavelength/spatial
crosscorrelation in each row is forced to ‘0’. A number getting repeated in
a row would mean the same code being assigned to two users having a common
wavelength. This would increase the maximum crosscorrelation between such users
to K, which is undesirable. Hence, the cardinality of this code family is less than
W
C
2
. Finally, the array [A] is formed by vertical concatenation of [A
i
]’s, whose
dimension is (W − 1) × W. The array has an arrangement as follows. The first
two rows have a difference of ‘1’ between elements of such pairs. The next two
rows have a difference of ‘2’ between elements and finally, the (W −1)th row has
a difference of W/2 between elements. For the example, the array [A] is shown
below.
54
3.5 RWOP-based 2D OCDMA Code Families
A =
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
1 2 3 4 5 6
2 3 4 5 6 1
1 3 2 4 0 0
3 5 4 6 0 0
1 4 2 5 3 6
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
.
The zeros indicate the discarded pairs. The discarded pairs ensure zero cross-
correlation of numbers (wavelengths or space channels) in all rows of the array.
The reason for discarding some pairs during the sorting process is to make sure
that no number is repeated in each row. A number repeated in a row would mean
the same 1D code being assigned to two users having a common wavelength. This
would increase the maximum crosscorrelation between such users to K, which is
undesirable. Hence, the cardinality of 2D code families employing this algorithm
is less than
W
C
2
. An advantage of this construction is the construction of code
families from any even numbers greater than 2. A similar approach can be used
for construction of row-wise orthogonal triples, but the cardinality of code families
reduces. Moreover, there are more efficient ways of constructing orthogonal triple
systems like BIBD and Steiner Triple Systems (STS), among others.
3.5 RWOP-based 2D OCDMA Code Families
The row-wise orthogonal pairs generated from the algorithm can be applied in the
construction of 2D wavelength-time and space-time code families. The application
of the algorithm to construct 2D wavelength-time code families of different weights
with λ
a
≤ 2 and λ
c
≤ 1 is explained in 3.5.1. The performance analysis and its
comparison with existing 2D code families is shown in 3.5.2.
55
OCDMA Code Families based on a Novel RWOP Algorithm
3.5.1 Construction of RWOP-based 2D Code Families
Each pair of wavelengths is assigned to a user with an OOC along both the wave-
lengths. Pairs from the same row are assigned a distinct 1D OOC (time spreading),
so that code reuse is possible and still have minimum crosscorrelation. This con-
struction ensures that, no two users are assigned the same code on the same pair of
wavelengths or a common wavelength. Each user is assigned a pair of wavelengths
carrying the same 1D OOC. The weight of these MPR code families is twice the
weight of the 1D OOC family used. The factor ‘2’ arises due to pair-wise allotment
of wavelengths. This construction limits the maximum crosscorrelation between
any two users to ‘1’ and has an out of phase autocorrelation of ‘0’ or ‘2’. Hence,
the proposed 2D code families are represented as (W ×T, K

, 2, 1).
The temporal length of the constructed code families is equal to the temporal
length of the 1D OOC employed. Since each row of the wavelength assignment
array is allotted a 1D code, the number of rows of the wavelength assignment array
decides the number of 1D codes to be used. Hence, the temporal length of these
2D code families is given by T ≥ K(K−1)(W−1) +1 as the number of 1D OOCs
used is W − 1. Table 3.5 shows an example of the way the proposed algorithm
is applied for 2D OCDMA code construction with ‘8’ wavelengths. The 1D OOC
shown in the right most column is constructed using BIBD with K = 3, W −1 =
7 and T = T
opt
= 43. In this 2D example, K

= 6, N
max
= 26, P

e
= 8.703 × 10
−8
,
where P

e
is the 2D probability of error due to MAI.
Similarly, a 1D OOC with K = 2, W − 1 = 7 and T = T
opt
= 15 can be
used as shown in Table 3.6. The resulting 2D construction has K

= 4, N
max
=
26, P

e
= 2.102 × 10
−4
. The numbers in Tables 3.5 and 3.6 represent wavelength
indices. These code families can also be used for 2D space-time OCDMA systems,
by replacing wavelength allocation with spatial allocation.
56
3.5 RWOP-based 2D OCDMA Code Families
Table 3.5: Example 2D code construction for W = 8, K

= 6
Wavelength Allocation 1D OOC
(w
1
, w
2
) (w
3
, w
4
) (w
5
, w
6
) (w
7
, w
8
) [1, 6, 36]
User 1 User 2 User 3 User 4
(w
2
, w
3
) (w
4
, w
5
) (w
6
, w
7
) (w
8
, w
1
) [3, 18, 22]
User 5 User 6 User 7 User 8
(w
1
, w
3
) (w
2
, w
4
) (w
5
, w
7
) (w
6
, w
8
) [9, 11, 23]
User 9 User 10 User 11 User 12
(w
3
, w
5
) (w
4
, w
6
) (w
7
, w
1
) (w
8
, w
2
) [26, 27, 33]
User 13 User 14 User 15 User 16
(w
1
, w
4
) (w
2
, w
5
) (w
3
, w
6
) [13, 35, 38]
User 17 User 18 User 19
(w
4
, w
7
) (w
5
, w
8
) (w
6
, w
1
) [19, 28, 39]
User 20 User 21 User 22
(w
1
, w
5
) (w
2
, w
6
) (w
3
, w
7
) (w
4
, w
8
) [14, 31, 41]
User 23 User 24 User 25 User 26
Table 3.6: Example 2D code construction for W = 8, K

= 4
Wavelength Allocation 1D OOC
(w
1
, w
2
) (w
3
, w
4
) (w
5
, w
6
) (w
7
, w
8
) C
1
= [1, 15]
(w
2
, w
3
) (w
4
, w
5
) (w
6
, w
7
) (w
8
, w
1
) C
2
= [2, 14]
(w
1
, w
3
) (w
2
, w
4
) (w
5
, w
7
) (w
6
, w
8
) C
3
= [3, 13]
(w
3
, w
5
) (w
4
, w
6
) (w
7
, w
1
) (w
8
, w
2
) C
4
= [4, 12]
(w
1
, w
4
) (w
2
, w
5
) (w
3
, w
6
) C
5
= [5, 11]
(w
4
, w
7
) (w
5
, w
8
) (w
6
, w
1
) C
6
= [6, 10]
(w
1
, w
5
) (w
2
, w
6
) (w
3
, w
7
) (w
4
, w
8
) C
7
= [7, 9]
3.5.2 Analysis of RWOP-based 2D Code Families
One of the performance metrics for 2D code families is cardinality versus optimum
code dimension. The optimum code dimension is a product of number of wave-
57
OCDMA Code Families based on a Novel RWOP Algorithm
lengths and the optimum temporal length of the constructed 2D code families.
The analysis of spectral efficiency of these 2D code families is shown in Sec. 3.6.2
along with the spectral efficiency of the RWOP-based 3D code families. The other
2D metric that has been taken into account is probability of error due to MAI ver-
sus number of active users. The above performance metrics are evaluated based on
the following assumptions: Optical fiber channel is a positive and additive system,
i.e., there is no error in detection when a ‘1’ is transmitted and the probability of
erroneous detection exists only when a ‘0’ is transmitted [26] [47] [54]. Receiver
has the knowledge of the code array of the user. All channels (wavelength and/or
space) are detected synchronously by the receiver equipped with an optical hard
limiter. Equal probability of transmission of ‘0’ and ‘1’ is assumed.
The temporal length of the 1D OOC family depends on the weight and number
of codes in the 1D OOC family. The number of codes in the 1D OOC family to be
used is decided by the RWOP algorithm and is given as (W−1). For (W−1) codes,
temporal length is found either by BIBD or by the method of extended sets. In
either case, the optimum temporal length is T
opt
= K(K−1)(W−1)+1. 1D OOC
families of weight K = 2, 3 have been used in this analysis, which give K

= 4, 6.
The cardinality of the 2D code families is the number of non-zero elements in the
generated array divided by two. Figure 3.2 shows a plot of the cardinality of 1D
OOCs and constructed 2D code families as a function of optimum code dimension.
For equal weights of 1D and 2D code families, the cardinality of 2D code families
is higher at equal code dimension.
The probability of error due to MAI of 2D code families is dependent on the
probability of error due to MAI of 1D OOC families (λ
c
≤ 1) being used. Reference
[26] gives the upper bound of probability of error due to MAI of 1D OOCs having a
crosscorrelation value of ‘1’, for any threshold value of detection. The probability
of error equation [26] for hard-limiting receiver, assuming chip synchronous case
58
3.5 RWOP-based 2D OCDMA Code Families
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0
200
400
600
800
1000
1200
1400
1600
C
a
r
d
i
n
a
l
i
t
y
Code Dimension
1D: K=4
1D: K=6
2D: K’=4
2D: K’=6
Figure 3.2: Cardinality of 1D OOCs
(K=4,6) and 2D RWOP-based
(K

=4,6) code families with λ
c
≤ 1 for
optimum code dimension
10 20 30 40 50 60 70 80 90 100 110
10
−8
10
−7
10
−6
10
−5
10
−4
Number of Active Users
P
r
o
b
a
b
i
l
i
t
y

o
f

E
r
r
o
r
W=8, T=15, K’=4, N
max
=26
W=10, T=19, K’=4, N
max
=39
W=12, T=23, K’=4, N
max
=60
W=14, T=27, K’=4, N
max
=79
W=16, T=31, K’=4, N
max
=108
Figure 3.3: Probability of error due to
MAI of the 2D RWOP-based code fam-
ilies having K

= 4, λ
a
≤ 2, λ
c
≤ 1
for different values of temporal length,
number of wavelengths, spatial chan-
nels and cardinality
for a threshold equal to the weight of the code is used to derive the probability of
error due to MAI in 2D and 3D code families. Without any loss of generality, the
upper bound of probability of error due to MAI for the designed 2D code families
is
P

e
=
P
e
(N
w
i
) ×P
e
_
N
w
j
_
T
. (3.5)
where, P
e
(N
w
i
) and P
e
(N
w
j
) represent the 1D probability of error for the pair of
wavelengths w
i
and w
j
respectively which are assigned to a user. We consider the
product of P
e
(N
w
i
) and P
e
(N
w
j
) because the probability of erroneous detection
is possible only if there is an erroneous detection of the code on both channels
simultaneously. The factor 1/T is attributed to simultaneous error detection. In
other words, there is a probability of error only when all bit positions on each
wavelength are detected erroneously at the same time (there should not be any
time difference between the erroneous detections on the two wavelengths). N
w
i
and N
w
j
refer to the number of interfering codes on wavelengths w
i
and w
i
respec-
tively. The wavelength allotment by the proposed code families is not uniform,
due to which the number of interfering wavelengths for the two wavelengths as-
59
OCDMA Code Families based on a Novel RWOP Algorithm
signed to a user differ, or, number of interfering codes for P
e
(N
w
i
) and P
e
(N
w
j
) are
different. The number of interfering codes in each wavelength is determined from
the generated array. The wavelength pairs allotted to all the users are determined.
Wavelengths assigned to each user are compared with the number of interfering
wavelengths to get the different number of interfering codes. The probability of
error of all the users is calculated and averaged. To obtain curves correspond-
ing to different number of active users, the probability of error is determined by
successively increasing the number of rows of the generated array.
An example to illustrate selection of system parameters is described in this
paragraph. For the example shown in Table 3.6, let us find the probability of
error due to MAI for a user with wavelengths (w
2
, w
4
) (corresponding to third
row, second column in Table 3.6). Interference for wavelength w
2
would be from
user (w
1
, w
2
) in first row, user (w
2
, w
3
) from second row and so on yielding N
w
2
= 5.
Similarly, interference for wavelength w
4
would yield N
w
4
= 6. The values of K
and T are 2 and 15 respectively for the example considered. The 1D P
e
is a
function of N, K and T as seen from Eqn. (2.6). So P

e
is the product of P
e
for
N
w
2
, K, T and P
e
for N
w
4
, K, T divided by T.
The probability of error for the 2D code families designed using RWOP for
weights K

= 4 and 6 is shown in Figs. 3.3 and 3.4 respectively. The 2D code
families with weights K

= 6 and K

= 4 have probability of error below 2 ×10
−7
and 3 ×10
−4
respectively.
Figure 3.5 shows the comparison of probability of error among (11×61, 6, 1, 1)
MWOOC, (13 × 156, 12, 0, 1) GMWRSC and the proposed 2D designs. (12 ×
67, 6, 2, 1), (18 × 35, 4, 2, 1) and (18 × 103, 6, 2, 1) are the proposed 2D examples
used for comparison. The probability of error for (12 × 67, 6, 2, 1) 2D design is
lower by a factor of about 10
−4
at 0.25 times the cardinality when compared with
(11 ×61, 6, 1, 1) MWOOC. The probability of error for (18 ×35, 4, 2, 1) 2D design
60
3.5 RWOP-based 2D OCDMA Code Families
10 20 30 40 50 60 70 80 90 100 110
10
−12
10
−11
10
−10
10
−9
10
−8
10
−7
Number of Active Users
P
r
o
b
a
b
i
l
i
t
y

o
f

E
r
r
o
r
W=8, T=43, K’=6, N
max
=26
W=10, T=55, K’=6, N
max
=39
W=12, T=67, K’=6, N
max
=60
W=14, T=79, K’=6, N
max
=79
W=16, T=91, K’=6, N
max
=108
Figure 3.4: Probability of error due to
MAI of the 2D RWOP-based code fam-
ilies having K

= 6, λ
a
≤ 2, λ
c
≤ 1
for different values of temporal length,
number of wavelengths, spatial chan-
nels and cardinality
20 40 60 80 100 120 140
10
−14
10
−12
10
−10
10
−8
10
−6
10
−4
10
−2
Number of Active Users
P
r
o
b
a
b
i
l
i
t
y

o
f

E
r
r
o
r
MWOOC:W=11, T=61, K’=6, N
max
=242
Proposed: W=18, T=35, K’=4, N
max
=135
Proposed: W=12, T=67, K’=6, N
max
=60
GMWRSC:W=13, T=156, K’=12, N
max
=169
Proposed:W=18, T=103, K’=6, N
max
=135
Figure 3.5: Comparison of (12 ×
67, 6, 2, 1), (18 ×35, 4, 2, 1) 2D RWOP-
based code with (13 × 67, 3, 1, 1)
MWOOC families and (18×103, 6, 2, 1)
2D RWOP-based code with (13 ×
156, 12, 0, 1) GMWRSC families
is lower by a factor of about 10
−2
, however at smaller cardinality (about 0.56
times) when compared with (11 ×61, 6, 1, 1) MWOOC. The comparison between
(13 × 156, 12, 0, 1) GMWRSC and (18 × 103, 6, 2, 1) 2D design shows that the
probability of error is lower by a factor of about 10
−4
for our code, but with
smaller cardinality (about 0.8 times). The probability of error of MWOOCs is
slightly lower when compared with GMWRSCs of equivalent code dimension. The
construction of GMWPCs, GMWRSCs and MWOOCs are based on primes and
product of primes. The design proposed here is based on even numbers. For
an OCDMA system, when the number of available wavelengths and /or spatial
channels are known, a code family can be constructed by choosing an even number
less than or equal to the available number.
Validation of the analytical results of the probability of error of the proposed
codes is shown in Fig. 3.6 by simulation. Simulation for 2D code family having
W = 6 was done using Matlab. Simulations validate the analysis given in Eqn.
(3.5) as the equations give the upper bound.
An example of an RWOP-based 2D construction for W = 4, T = 7 and K

= 4
61
OCDMA Code Families based on a Novel RWOP Algorithm
7 8 9 10 11 12 13 14
10
−6
10
−5
10
−4
Number of Active Users
P
r
o
b
a
b
i
l
i
t
y

o
f

E
r
r
o
r
Simulation: W=6, T=11, N
max
=13
Theory: W=6, T=11, N
max
=13
Figure 3.6: Validation of analytical results of probability of error due to MAI of
a 2D RWOP-based code family having W = 6, T = 11, K

= 4, N
max
= 13 by
simulation
is shown in Table 3.7 in the same representation as the GMWPC, GMWRSC,
MWOOC and FOOC families’ examples. This construction uses three 1D OOCs
having K = 2 and T = 7. The wavelengths are represented as w
1
, w
2
, w
3
and w
4
while the time chips are shown as t
1
, t
2
, . . . , t
7
. This code family is compared with
other 2D code families reported in literature (examples shown in Tables 3.1, 3.2,
3.3 and 3.4) as shown in Table 3.8.
Table 3.7: Example 2D RWOP-based code construction for W = 4, T = 7 &
K

= 4
Users W : T
U
1
(w
1
: t
1
t
7
), (w
2
: t
1
t
7
)
U
2
(w
3
: t
1
t
7
), (w
4
: t
1
t
7
)
U
3
(w
2
: t
2
t
6
), (w
3
: t
2
t
6
)
U
4
(w
4
: t
2
t
6
), (w
1
: t
2
t
6
)
U
5
(w
1
: t
3
t
5
), (w
3
: t
3
t
5
)
U
6
(w
2
: t
3
t
5
), (w
4
: t
3
t
5
)
The probability of error due to MAI as shown in Table 3.8 is for maximum
possible number of active users of the respective code family. It should be noted
that the code dimension of all the code families vary from 27 to 30. This in turn
indicates that the code families with higher cardinality and lower weight lead to
high probability of error due to MAI. It may be noted that among GMWPC,
62
3.6 RWOP-based 3D OCDMA Code Families
FOOC and RWOP code families, code dimension, code weight and cardinality are
equivalent, and the probability of error for the RWOP-based 2D code family is
lower than that of the other two code families.
Table 3.8: Comparison of example 2D code families
Code Family Code Dimension Code Weight Cardinality P
e
(W ×T) (K

) (N
max
) (MAI)
GMWPC 4 ×7 4 7 1.10 ×10
−3
GMWRSC 3 ×10 2 15 3.88 ×10
−2
MWOOC 3 ×9 2 36 2.55 ×10
−1
FOOC 3 ×9 4 6 1.04 ×10
−2
RWOP 4 ×7 4 6 5.95 ×10
−5
3.6 RWOP-based 3D OCDMA Code Families
The row-wise orthogonal pairs are also applied to construct 3D space-wavelength-
time code families. Unlike the 3D SPP and MPP code families reported in [102],
these 3D MPP code families have a higher out-of-phase autocorrelation value.
The reported 3D code families are constructed by using 2D GMWPC families,
while our constructions are based on the RWOP algorithm. The construction
of K
′′
= 8, 12 3D code families with S = W and S < W is explained in 3.6.1.
The performance analysis and comparison of the constructed 3D code families are
shown in 3.6.2.
3.6.1 Construction of RWOP-based 3D Code Families
The RWOP algorithm is used for both wavelength as well as spatial allocation.
Each pair of spatial channels is paired with all the wavelength pairs of the cor-
responding row. As in the case of 2D, each row is assigned a distinct 1D OOC
63
OCDMA Code Families based on a Novel RWOP Algorithm
spread in the time domain. This design uses equal number of wavelengths and
spatial channels. The construction also ensures a maximum crosscorrelation value
of ‘1’ between any two users. Each user is allotted a pair of spatial channels each
carrying a pair of wavelengths. Along both the spatial channels, all wavelengths
of a particular user carry the same 1D OOC simultaneously. The weight of these
MPP code families is four times the weight of the 1D OOC family employed. The
factor ‘4’ arises due to the simultaneous transmission of the 1D OOC along four
channels (‘2’ spatial channels, each using ‘2’ wavelengths for a user). Hence, these
3D code families are represented as (S ×W ×T, K
′′
, 4, 1).
Using this construction, the temporal length of the code families is given by
T
opt
= K(K − 1)(W − 1) + 1. Table 3.9 shows an example of 3D OCDMA code
family constructed for 8 wavelengths and 8 spatial channels. The 1D OOC shown
in the right most column has K = 2, W − 1 = 7 and T = T
opt
= 15. In this 3D
example, K
′′
= 8, N
max
= 98, P
′′
e
= 1.315 × 10
−8
, where P
′′
e
is the 3D probability
of error due to MAI.
Table 3.9: Example 3D code construction for S = W = 8, K
′′
= 8
Spatial Allocation Wavelength Allocation 1D OOC
(s
1
, s
2
)(s
3
, s
4
)(s
5
, s
6
)(s
7
, s
8
) (w
1
, w
2
)(w
3
, w
4
)(w
5
, w
6
)(w
7
, w
8
) C
1
= [1, 15]
(s
2
, s
3
)(s
4
, s
5
)(s
6
, s
7
)(s
8
, s
1
) (w
2
, w
3
)(w
4
, w
5
)(w
6
, w
7
)(w
8
, w
1
) C
2
= [2, 14]
(s
1
, s
3
)(s
2
, s
4
)(s
5
, s
7
)(s
6
, s
8
) (w
1
, w
3
)(w
2
, w
4
)(w
5
, w
7
)(w
6
, w
8
) C
3
= [3, 13]
(s
3
, s
5
)(s
4
, s
6
)(s
7
, s
1
)(s
8
, s
2
) (w
3
, w
5
)(w
4
, w
6
)(w
7
, w
1
)(w
8
, w
2
) C
4
= [4, 12]
(s
1
, s
4
)(s
2
, s
5
)(s
3
, s
6
) (w
1
, w
4
)(w
2
, w
5
)(w
3
, w
6
) C
5
= [5, 11]
(s
4
, s
7
)(s
5
, s
8
)(s
6
, s
1
) (w
4
, w
7
)(w
5
, w
8
)(w
6
, w
1
) C
6
= [6, 10]
(s
1
, s
5
)(s
2
, s
6
)(s
3
, s
7
)(s
4
, s
8
) (w
1
, w
5
)(w
2
, w
6
)(w
3
, w
7
)(w
4
, w
8
) C
7
= [7, 9]
Another way of employing the proposed algorithm to construct 3D code fam-
ilies is shown in Table 3.10. As shown in the example, the fiber allocation is for
S = 4 and the wavelength allocation is for W = 8. The array for S = 4, which has
three rows is repeated and truncated so that the number of rows of fiber allocation
and wavelength allocation are same. The allocation of fibers, wavelengths and 1D
64
3.6 RWOP-based 3D OCDMA Code Families
OOC is similar to that shown in the example for S = W = 6. This design gives
the flexibility of choosing an even number of fibers, much less than the number of
wavelengths.
Table 3.10: Example 3D code construction for S = 4, W = 8, K
′′
= 8
Spatial Alloc. Wavelength Allocation 1D OOC
(s
1
, s
2
) (s
3
, s
4
) (w
1
, w
2
) (w
3
, w
4
) (w
5
, w
6
) (w
7
, w
8
) C
1
= [1, 6, 36]
(s
2
, s
3
) (s
4
, s
1
) (w
2
, w
3
) (w
4
, w
5
) (w
6
, w
7
) (w
8
, w
1
) C
2
= [3, 18, 22]
(s
1
, s
3
) (s
2
, s
4
) (w
1
, w
3
) (w
2
, w
4
) (w
5
, w
7
) (w
6
, w
8
) C
3
= [9, 11, 23]
(s
1
, s
2
) (s
3
, s
4
) (w
3
, w
5
) (w
4
, w
6
) (w
7
, w
1
) (w
8
, w
2
) C
4
= [26, 27, 33]
(s
2
, s
3
) (s
4
, s
1
) (w
1
, w
4
) (w
2
, w
5
) (w
3
, w
6
) C
5
= [13, 35, 38]
(s
1
, s
3
) (s
2
, s
4
) (w
4
, w
7
) (w
5
, w
8
) (w
6
, w
1
) C
6
= [19, 28, 39]
(s
1
, s
2
) (s
3
, s
4
) (w
1
, w
5
) (w
2
, w
6
) (w
3
, w
7
) (w
4
, w
8
) C
7
= [14, 31, 41]
3.6.2 Analysis of RWOP-based 3D Code Families
For 3D code families, the non-zero elements of every row are counted and the
resultant number is divided by two. These numbers for every row are first squared
and then added to get the cardinality of the 3D code families designed by using
the proposed algorithm. Figure 3.7 shows a plot of the cardinality of 1D OOCs
and constructed 3D code families as a function of optimum code dimension. For
equal weights of 1D and 3D code families, the cardinality of 3D code families is
higher at equal code dimension. The spectral efficiency of the constructed 2D and
3D code families is shown in Fig. 3.8. The spectral efficiency of the constructed
2D code families is given by
η(2D) =
N
max
(2D)
W[K(K −1)(W −1) + 1]
. (3.6)
65
OCDMA Code Families based on a Novel RWOP Algorithm
Since N
max
(2D) ≤ W(W −1)/2, the simplified spectral efficiency is given by
η(2D) ≤ 1/ [2K(K −1) + 2/(W −1)] . (3.7)
For K = 2 ⇒ K

= 4, η(2D) ≤ 1/ [4 + 2/(W −1)] and would approach 0.25
for large W. The actual spectral efficiency of the constructed weight 4 2D code
families is found to be between 0.2 to 0.22. In case of GMWRSCs of weight 4,
the spectral efficiency is 0.25. For K = 3 ⇒K

= 6, η(2D) ≤ 1/ [12 + 2/(W −1)]
and would approach 0.083 for large W. The actual spectral efficiency of the
constructed weight 6 2D code families is found to be between 0.07 to 0.08. In case
of GMWRSCs of weight 6, the spectral efficiency is 0.16.
The spectral efficiency of the constructed 3D code families with S = W is
given by
η(3D) =
N
max
(3D)
SW[K(K −1)(W −1) + 1]
. (3.8)
Since N
max
(3D) ≤ (W/2)
2
(W − 1), the simplified spectral efficiency is given
by
η(3D) ≤ 1/ [4K(K −1) + 4/(W −1)] . (3.9)
For K = 2 ⇒K
′′
= 8, η(3D) ≤ 1/ [8 + 4/(W −1)] and would approach 0.125
for large W. The actual spectral efficiency of the constructed weight 8 3D code
families with S = W is found to be between 0.09 to 0.11. For equivalent SPP code
families, the spectral efficiency is 1 for S = W, greater than 1 for S < W and 0.125
for S > W, K
′′
= 8. For K = 3 ⇒ K
′′
= 12, η(3D) ≤ 1/ [24 + 4/(W −1)] and
would approach 0.041 for large W. The actual spectral efficiency of the constructed
weight 12 3D code families with S = W is found to be between 0.03 to 0.04. For
equivalent SPP code families, the spectral efficiency is 0.08 for S > W, K
′′
= 12.
66
3.6 RWOP-based 3D OCDMA Code Families
The actual spectral efficiency of 3D code families with S < W is better than the
S = W 3D code families in most cases as the allocation is optimum for S = 4.
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0
100
200
300
400
500
600
700
800
C
a
r
d
i
n
a
l
i
t
y
Code Dimension
1D: K=8
1D: K=12
3D (S=W): K’’=8
3D (S=W): K’’=12
3D (S<W): K’’=8
3D (S<W): K’’=12
Figure 3.7: Cardinality of 1D OOCs
(K=8,12) and 3D RWOP-based
(K
′′
=8,12) code families with λ
c
≤ 1
for optimum code dimension
0 2000 4000 6000 8000 10000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
S
p
e
c
t
r
a
l

E
f
f
i
c
i
e
n
c
y
Code Dimension
2D: K’=4
2D: K’=6
3D (S=W): K’’=8
3D: K’’=12
3D (S<W): K’’=8
3D (S<W): K’’=12
Figure 3.8: Spectral efficiency of 2D
(K=4,6) and 3D (K
′′
=8,12) OCDMA
code families constructed using RWOP
algorithm
The upper bound of probability of error due to MAI for the constructed 3D
code families is
P
′′
e
=
(P
e
(N
s
i
,w
i
))
_
P
e
(N
s
i
,w
j
)
_ _
P
e
(N
s
j
,w
i
)
_ _
P
e
(N
s
j
,w
j
)
_
T
3
(3.10)
where, P
e
(N
s
i
,w
i
), P
e
(N
s
i
,w
j
), P
e
(N
s
j
,w
i
) and P
e
(N
s
j
,w
j
), represent the 1D prob-
ability of error for the pair of spatial channels s
i
and s
j
carrying the pair of
wavelengths w
i
and w
j
, which are assigned to a user. The spatial channel and
wavelength allotment by the proposed code design is not uniform, due to which
the number of spatial channels and the interfering wavelengths in each spatial
channel for the four channels assigned to a user differ. Since the same code is used
for both spatial channel allocation and wavelength allocation, the number of pairs
of spatial channels and wavelengths are equal. These pairs and their number are
determined. For each pair of spatial channels allotted to a user, the overlapping
67
OCDMA Code Families based on a Novel RWOP Algorithm
spatial channels of other users are found. For each spatial channel of a user’s
spatial channel pair, the rows of the array which have overlapping spatial chan-
nels are formed into a sub-array. The number of interfering wavelengths for each
spatial channel pair of a user is determined. The maximum number of interfering
wavelengths is considered to obtain the probability of error. The curves of prob-
ability of error due to MAI for different numbers of active users are plotted by
successively increasing the number of rows of the generated array. The probability
of error is zero when less than ‘3’ rows of the array are used for the 3D OCDMA
systems.
The probability of error for the 3D code families designed using RWOP for
weights K

= 8 and 12 is shown in Figs. 3.9 and 3.10 respectively. The design
gives probability of error due to MAI below 10
−14
and 1.5 ×10
−8
for code families
with weights K
′′
= 12 and K
′′
= 8 respectively. The small dip in the curves
of Figs. 3.9 and 3.10 may be attributed to uneven allocation in various rows.
For example, the rows in the middle of the generated array have less number of
elements. The curves follow the conventional increase in probability of error with
increasing number of active users. The lowest points of each individual curve
indicate that the probability of error is zero for number of active users less than
that corresponding to the lowest points. This shows the probability of error due
to MAI is zero even when the number of active users is greater than the weight of
the code. This is due to zero crosscorrelation among wavelengths and/ or spatial
channels of different users in each row. The probability of error for the 3D code
families with unequal number of spatial channels and wavelengths is shown in
Fig. 3.11. Figure 3.12 shows the comparison of proposed 3D code families for
S = W designs and S < W designs. It can be seen that the bit error rate and the
cardinality are almost same for equivalent code dimension.
Figure 3.13 shows the probability of error due to MAI for (16×3×127, 16, 0, 1),
68
3.6 RWOP-based 3D OCDMA Code Families
0 100 200 300 400 500 600 700 800
10
−16
10
−14
10
−12
10
−10
10
−8
Number of Active Users
P
r
o
b
a
b
i
l
i
t
y

o
f

E
r
r
o
r
S=6,W=6,T=11,K’’=8,N
max
=35
S=8,W=8,T=15,K’’=8,N
max
=98
S=10,W=10,T=19,K’’=8,N
max
=171
S=12,W=12,T=23,K’’=8,N
max
=334
S=14,W=14,T=27,K’’=8,N
max
=485
S=16,W=16,T=31,K’’=8,N
max
=788
Figure 3.9: Probability of error due to
MAI of the 3D RWOP-based code fam-
ilies having K
′′
= 8, λ
a
≤ 4, λ
c
≤ 1
for different values of temporal length,
number of wavelengths, spatial chan-
nels and cardinality
0 100 200 300 400 500 600 700 800
10
−26
10
−24
10
−22
10
−20
10
−18
10
−16
10
−14
Number of Active Users
P
r
o
b
a
b
i
l
i
t
y

o
f

E
r
r
o
r
S=6,W=6,T=31,K’’=12,N
max
=35
S=8,W=8,T=43,K’’=12,N
max
=98
S=10,W=10,T=55,K’’=12,N
max
=171
S=12,W=12,T=67,K’’=12,N
max
=334
S=14,W=14,T=79,K’’=12,N
max
=485
S=16,W=16,T=91,K’’=12,N
max
=788
Figure 3.10: Probability of error due to
MAI of the 3D RWOP-based code fam-
ilies having K
′′
= 12, λ
a
≤ 4, λ
c
≤ 1
for different values of temporal length,
number of wavelengths, spatial chan-
nels and cardinality
0 500 1000 1500 2000 2500 3000
10
−35
10
−30
10
−25
10
−20
10
−15
Number of Simultaneous Users
B
i
t

E
r
r
o
r

R
a
t
e
S=4, W=30, T=175, C=770
S=6, W=40, T=235, C=1792
S=6, W=50, T=295, C=2809
Figure 3.11: Probability of error due to
MAI of the S < W 3D RWOP-based
code families with K
′′
= 12, λ
a
≤ 4,
λ
c
≤ 1
100 200 300 400 500 600 700 800 900 1000 1100
10
−30
10
−25
10
−20
10
−15
Number of Simultaneous Users
B
i
t

E
r
r
o
r

R
a
t
e
S=W=18, T=103, K’’=12, C=1087
S=10, W=24, T=139, K’’=12, C=1077
S=12, W=22, T=127, K’’=12, C=1099
Figure 3.12: Comparison of probability
of error due to MAI between S < W
and S = W 3D RWOP-based code fam-
ilies
(16 × 4 × 127, 16, 0, 1) prime based SPP 3D code families and the (10 × 10 ×
55, 12, 4, 1), (14 ×14 ×27, 8, 4, 1), (16 ×16 ×31, 8, 4, 1) proposed 3D code families.
It can be noted from Fig. 3.13, that probability of error for the (10×10×55, 12, 4, 1)
proposed 3D code family is less than that of the (16 ×3 ×127, 16, 0, 1) SPP code
family with 0.45 times the corresponding cardinality. The probability of error
for the proposed (14 × 14 × 27, 8, 4, 1) 3D code family is lower than that of the
69
OCDMA Code Families based on a Novel RWOP Algorithm
(16 ×3 ×127, 16, 0, 1) SPP code family and has a higher cardinality of about 1.27
times. The probability of error for the proposed (16 × 16 × 31, 8, 4, 1) 3D code
family is lower than that of the (16 ×4 ×127, 16, 0, 1) SPP code family with 1.55
times the corresponding cardinality. The probability of error characteristics of the
3D code families have a lower slope as compared to those of the SPP code families.
0 100 200 300 400 500 600 700 800
10
−25
10
−20
10
−15
10
−10
10
−5
Number of Active Users
P
r
o
b
a
b
i
l
i
t
y

o
f

E
r
r
o
r
SPP(prime):S=16,W=3,T=127,K’’=16,N
max
=381
Proposed:S=10,W=10,T=55,K’’=12,N
max
=171
Proposed:S=14,W=14,T=27,K’’=8,N
max
=485
SPP(prime):S=16,W=4,T=127,K’’=16,N
max
=508
Proposed:S=16,W=16,T=31,K’’=8,N
max
=788
Figure 3.13: Comparison of SPP code
families with 3D RWOP-based code
families
10 20 30 40 50 60 70 80
10
−12
10
−10
10
−8
10
−6
10
−4
Number of Active Users
P
r
o
b
a
b
i
l
i
t
y

o
f

E
r
r
o
r
3−D: S=6, W=6, T=11, K’’=8, N
max
=35
2−D: W=8, T=43, K’=6, N
max
=26
2−D: W=14, T=27, K’=4, N
max
=79
Figure 3.14: Comparison of 2D RWOP-
based code families and 3D RWOP-
based code families
Fig. 3.14 shows a comparison of probability of error of the (6 ×6 ×11, 8, 4, 1)
3D RWOP-based code family with that of the (14 × 27, 4, 2, 1), (8 × 43, 6, 2, 1)
2D RWOP-based code families. The probability of error of the 3D code family
is lower than that of the 2D code families, but the 2D code family with weight
K

= 4 has a higher cardinality.
3.7 Summary
In this chapter, the crosscorrelation value between any two codes of the proposed
RWOP-based 2D and 3D code families is at most 1. Out-of-phase autocorrelation
value is a maximum of 2 for 2D code families and 4 for 3D code families. The
probability of error due to MAI of these code families is lower than that of pre-
viously reported code families. A comparison among different 2D code families
70
3.7 Summary
with the same code dimension shows lower probability of error due to MAI of the
RWOP-based code family. The spectral efficiency of the lower weight (2D: K

= 4
and 3D: K
′′
= 8) RWOP-based code families is comparable to previously reported
2D and 3D code families. For higher weight codes (2D: K

= 6 and 3D: K
′′
= 12),
the spectral efficiency is observed to be lower for RWOP-based code families.
Limitations of the RWOP algorithm can be summarised as; it can only be
used to construct code families based on even numbers. It means, the number
of wavelengths and/or space channels have to be even. All the rows of the array
generated by the RWOP algorithm are not complete, which is indicated by 0’s in
the example array A in Sec. 3.4. Some pairs of some rows in the array generated by
the RWOP algorithm are discarded so as to preserve the crosscorrelation property
of λ
c
≤ 1.
71
C H A P T E R 4
OCDMA Code Families based on
a Novel CRWOP algorithm
4.1 Introduction
This chapter deals with the construction and performance analysis of 2D and 3D
OCDMA code families based on an algorithm, which is an enhancement over the
RWOP algorithm and is named as complete row-wise orthogonal pairs (CRWOP).
The CRWOP algorithm is used for wavelength and/or spatial allocation and 1D
OOCs are used for time chip allocation to construct 2D wavelength-time or space-
time multipulse per row (MPR) and 3D space-wavelength-time multipulse per
plane (MPP) code families. The allocation of wavelengths and/or spatial channels
by the CRWOP algorithm is based on the concept of a dual-tone multi-frequency
(DTMF) signalling grid and is explained with an example. The construction of
2D wavelength-time and 3D space-wavelength-time code families are illustrated
with the help of examples.
OCDMA Code Families based on a Novel CRWOP algorithm
The analyzed performance metrics are cardinality, spectral efficiency and prob-
ability of error due to multiple access interference (MAI) for equivalent code di-
mension, which are the same as those for the RWOP-based 2D and 3D code
families (Chapter 3). The performance of these CRWOP-based code families is
compared with the RWOP-based code families and other reported literature. The
probability of error due to MAI of these code families is compared with other code
families based on equivalent cardinality and equivalent probability of error due to
MAI when all users are interfering. The probability of error due to MAI is also
compared based on equivalent code dimension.
Section 4.2 discusses the limitations of the RWOP algorithm and the CRWOP
algorithm is explained in Sec. 4.3. The construction and performance analysis
of 2D wavelength-time MPR and 3D space-wavelength-time MPP code families is
shown in Sections 4.4 and 4.5 respectively. Finally, Sec. 4.6 gives the summary of
this chapter.
4.2 Scope of Improvement in RWOP Algorithm
Code families constructed using the RWOP algorithm have an even number of
wavelengths and/or spatial channels. This constraint of the RWOP algorithm can
be improved by an algorithm which can construct code families from any given
number of wavelengths and/or spatial channels.
Since each pair in the array, generated by the RWOP algorithm represents the
wavelength/spatial allocation of a user in 2D wavelength-time/space-time code
families, the cardinality of the 2D code families is reduced due to the zeros in the
generated array. The cardinality of the 2D code families is less than W(W − 1)
due to the discarded pairs. This can be termed as the efficiency of allocating pairs
in a row, which is less than ‘1’.
74
4.3 The CRWOP Algorithm
In case of 3D code families, the effect of discarded pairs is even more pro-
nounced as the cardinality is the sum of product of number of pairs in a row. As
the efficiency of allocating pairs in a row is less than ‘1’, the product of the number
of pairs in a row is even lesser. The product is due to the same algorithm being
used for both spatial allocation as well as wavelength allocation as shown in Table
3.9.
For S < W, S = 4 3D code families, there are no discarded pairs in the spatial
allocation and the cardinality of the resulting 3D code families is higher than that
of S = W, S > 4 3D code families for equivalent code dimension. The reduced
cardinality leads to lower spectral efficiency of the RWOP-based 2D and 3D code
families. These limitations of the RWOP algorithm are overcome by the CRWOP
algorithm given below.
4.3 The CRWOP Algorithm
In this section, we propose a modified version of the RWOP algorithm which is
named here as CRWOP algorithm for the sake of interpretation of results. The
CRWOP algorithm is shown in the form of a flowchart in Fig. 4.1. The number of
wavelengths (W > 2) to be used in the system is the first parameter to be chosen.
The number of 1D OOCs to be used in time domain, n is a function of W. The
W wavelengths are arranged in a manner similar to the frequencies in a DTMF
signalling grid. The first row contains wavelengths (w
1
, w
2
, . . . , w
n
) and the first
column has wavelengths (w
n+1
, . . . , w
W
), where n = ⌈W/2⌉.
The development of the CRWOP algorithm is explained with the help of an
example giving each step of the algorithm. Choosing W = 7, the value of n is
calculated to be 4. A 2D grid of wavelengths is formed as shown in Table 4.1.
Each 1D OOC (C
1
, C
2
, . . . , C
n
) is allocated to each box of the empty wavelength
75
OCDMA Code Families based on a Novel CRWOP algorithm
START
W= no. of wavelengths
n =
_
W
2
_
columns←1 : n
rows←n + 1 : W
Sort - U
N
← (w
i
, w
j
)C
k
;
N = 1, 2, . . . N
max
,
N
max
= n ×(W −n),
k = 1, 2, . . . n.
Sort for ascending k
Club (w
i
, w
j
) pairs for same C
k
STOP
Figure 4.1: Flowchart depicting the proposed CRWOP algorithm
grid (Table 4.1) corresponding to pairs of wavelengths. The manner of 1D OOC
allocation is by a cyclic shift of the 1D OOCs in successive rows as shown in Table
4.2.
Table 4.1: Example of empty wave-
length grid for W = 7
w
1
w
2
w
3
w
4
w
5
w
6
w
7
Table 4.2: Example of allocated 1D
OOCs in wavelength grid for W = 7
w
1
w
2
w
3
w
4
w
5
C
1
C
2
C
3
C
4
w
6
C
2
C
3
C
4
C
1
w
7
C
3
C
4
C
1
C
2
Based on the allocation of 1D OOCs to wavelength pairs, the next step creates
an array according to wavelength pairs as shown in Table 4.3. In the next step,
76
4.3 The CRWOP Algorithm
the array formed in the previous step is sorted based on the ascending order of
users’ 1D OOCs as shown in Table 4.4.
Table 4.3: Example of created array based
on wavelength pairs for W = 7
(w
1
, w
5
)C
1
(w
2
, w
5
)C
2
(w
3
, w
5
)C
3
(w
4
, w
5
)C
4
(w
1
, w
6
)C
2
(w
2
, w
6
)C
3
(w
3
, w
6
)C
4
(w
4
, w
6
)C
1
(w
1
, w
7
)C
3
(w
2
, w
7
)C
4
(w
3
, w
7
)C
1
(w
4
, w
7
)C
2
Table 4.4: Example of array sorted
according to users’ 1D OOCs for
W = 7
U
1
←(w
1
, w
5
)C
1
U
2
←(w
4
, w
6
)C
1
U
3
←(w
3
, w
7
)C
1
U
4
←(w
2
, w
5
)C
2
U
5
←(w
1
, w
6
)C
2
U
6
←(w
4
, w
7
)C
2
U
7
←(w
3
, w
5
)C
3
U
8
←(w
2
, w
6
)C
3
U
9
←(w
1
, w
7
)C
3
U
1
0 ←(w
4
, w
5
)C
4
U
1
1 ←(w
3
, w
6
)C
4
U
1
2 ←(w
2
, w
7
)C
4
Clubbing the wavelength pairs corresponding to same 1D OOC results in the
formation of complete row-wise orthogonal pairs. The array generated by the CR-
WOP algorithm is a pair based design of dimension n×(W−n) with a wavelength
crosscorrelation of zero in each row. The completeness of the CRWOP algorithm
refers to the complete allotment of wavelengths in the row-wise orthogonal pairs
of all the rows. For the example of W = 7, the final generated array is shown in
Table 4.5.
Table 4.5: Example of CRWOP generated array for W = 7
(w
1
, w
5
) (w
4
, w
6
) (w
3
, w
7
) C
1
(w
2
, w
5
) (w
1
, w
6
) (w
4
, w
7
) C
2
(w
3
, w
5
) (w
2
, w
6
) (w
1
, w
7
) C
3
(w
4
, w
5
) (w
3
, w
6
) (w
2
, w
7
) C
4
Advantages of CRWOP algorithm over RWOP algorithm are inherent in its
ability of wavelength allotment with any number as compared to wavelength al-
77
OCDMA Code Families based on a Novel CRWOP algorithm
lotment from even numbers, complete allotment as compared to some dropped
pairs and lower temporal length of CRWOP-based multi-dimensional OCDMA
(MD-OCDMA) code families. This helps in accommodating larger number of
users for the same code dimension when compared to RWOP-based MD-OCDMA
code families.
4.4 CRWOP-based 2D OCDMA Code Families
The pairs generated from the CRWOP algorithm can be applied to construct 2D
wavelength-time and space-time code families. The construction of 2D wavelength-
time code families using the CRWOP algorithm for different weights with λ
a
≤ 2
and λ
c
≤ 1 is explained in 4.4.1. This construction gives an improvement in
cardinality for equivalent code dimension in comparison with the RWOP-based 2D
code families. The performance analysis and a comparative study with existing
2D code families is shown in 4.4.2.
4.4.1 Construction of CRWOP-based 2D Code Families
The generated array of the CRWOP algorithm is similar in structure to the array
generated from the RWOP algorithm. The wavelength allocation of the CRWOP-
based 2D code families is done in the same way as was done for the RWOP-
based 2D code families (Sec. 3.5.1). These code families also have maximum
crosscorrelation and out-of-phase autocorrelation values of 1 and 2 respectively.
In these constructions, the required number of 1D OOCs is given by n = ⌈W/2⌉.
So the optimum temporal length of these 2D code families is given by T
opt
=
K(K − 1)n + 1, where K is the weight of the employed 1D OOC family. The
weight of these constructed 2D code families is given by K

= 2K. An example
2D code family constructed using 9 wavelengths is shown in Table 4.6. The other
78
4.4 CRWOP-based 2D OCDMA Code Families
parameters in the example for W = 9 are n = 5, K = 2, K

= 4 and T = T
opt
= 11.
The code family shown in the example can be represented as (9 ×11, 4, 2, 1), with
N
max
= 20.
Table 4.6: Example 2D code construction for W = 9, K

= 4
Wavelength Allocation 1D OOC
(w
1
, w
6
) (w
5
, w
7
) (w
4
, w
8
) (w
3
, w
9
) C
1
= [1, 11]
(w
2
, w
6
) (w
1
, w
7
) (w
5
, w
8
) (w
4
, w
9
) C
2
= [2, 10]
(w
3
, w
6
) (w
2
, w
7
) (w
1
, w
8
) (w
5
, w
9
) C
3
= [3, 9]
(w
4
, w
6
) (w
3
, w
7
) (w
2
, w
8
) (w
1
, w
9
) C
4
= [4, 8]
(w
5
, w
6
) (w
4
, w
7
) (w
3
, w
8
) (w
2
, w
9
) C
5
= [5, 7]
A simpler form of showing the example in Table 4.6 can be of the form given
in [121]. The example of Table 4.6 is shown in the simpler form in Table 4.7. The
elements w
1
, w
2
, . . . , w
9
in the first column of Table 4.7 represent the wavelengths
and C
1
, C
2
, C
3
, C
4
, C
5
are the codes of a (11, 2, 1, 1) 1D OOC family employed
to construct the 2D code family shown in Tables 4.6 and 4.7. Similarly, (W ×
T, K

, 2, 1) 2D code families can be constructed for different values of W, T and
K

. As an alternative construction, the number of wavelengths can be chosen
depending on the required cardinality of an OCDMA system. To construct a 2D
code family with a cardinality greater than or equal to N
g
, the required number
of wavelengths is W =
__
4N
g
+ 1
_
.
4.4.2 Analysis of CRWOP-based 2D Code Families
The performance criteria of these code families includes cardinality and spectral
efficiency for varying code dimension as well as the probability of error due to
MAI for different numbers of active users. The cardinality of these code families
is N
max
= n(W − n) and the spectral efficiency is η(2D) =
n(W−n)
W(K(K−1)n+1)
. The
cardinality of the CRWOP-based 2D code families, the RWOP-based 2D code
families (Sec. 3.5.2) and previously reported 2D code families is shown in Fig.
79
OCDMA Code Families based on a Novel CRWOP algorithm
Table 4.7: Simpler form (same example as Table 4.6)
Each column is a code of the (9 ×11, 4, 2, 1) 2D code family
w
1
C
1
C
2
C
3
C
4
w
2
C
2
C
3
C
4
C
5
w
3
C
1
C
3
C
4
C
5
w
4
C
1
C
2
C
4
C
5
w
5
C
1
C
2
C
3
C
5
w
6
C
1
C
2
C
3
C
4
C
5
w
7
C
1
C
2
C
3
C
4
C
5
w
8
C
1
C
2
C
3
C
4
C
5
w
9
C
1
C
2
C
3
C
4
C
5
4.2. From curves (a) and (b) of Fig. 4.2, we can see that the cardinality of the
CRWOP-based 2D code families is higher than that of RWOP-based 2D code
families. Curves (c), (d), (e) and (f) show the cardinality of MWOOCs to increase
sharply with the code dimension and is higher than that of CRWOP-based 2D
code families for low code weights (K

= 4, 7, 10). For a large code weight of 12
(required for low probability of error due to MAI), the cardinality of MWOOC is
lower than that of CRWOP-based and RWOP-based 2D code families.
0 1000 2000 3000 4000 5000 6000 7000 8000
0
200
400
600
800
1000
1200
1400
1600
1800
C
a
r
d
i
n
a
l
i
t
y
Code Dimension
(a) RWOP−2D: K’=4
(b) CRWOP−2D: K’=4
(c) MWOOC−2D: K’=4
(d) MWOOC−2D: K’=7
(e) MWOOC−2D: K’=10
(f) MWOOC−2D: K’=12
Figure 4.2: Cardinality of 2D CRWOP-
based, RWOP-based and MWOOC
OCDMA code families
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
p
e
c
t
r
a
l

E
f
f
i
c
i
e
n
c
y
Code Dimension
RWOP: K’=4
CRWOP: K’=4
MWOOC: K’=10
MWOOC: K’=12
Figure 4.3: Spectral efficiency of 2D
CRWOP-based, RWOP-based and
MWOOC OCDMA code families
Figure 4.3 shows the spectral efficiency of weight 4 CRWOP-based and RWOP-
based 2D code families and MWOOC 2D code families of weight 10 and 12. The
80
4.4 CRWOP-based 2D OCDMA Code Families
spectral efficiency of the CRWOP-based 2D code families is higher than that of
RWOP-based 2D code families. The spectral efficiency of the CRWOP-based 2D
code families is lower than that of weight 10 MWOOC 2D code families and higher
than that of weight 12 MWOOC 2D code families. The cardinality and spectral
efficiency of the MWOOC 2D code families with large weight (K

= 10, 12) have
been compared with K

= 4 CRWOP-based 2D code families owing to the large
probability of error due to MAI of the MWOOC 2D code families for small weights
as shown in Fig. 4.5.
The probability of error due to MAI corresponding to wavelengths w
i
and w
j
are represented as P
e
(N
w
i
) and P
e
_
N
w
j
_
. Since the two 1D OOCs along wave-
lengths w
i
and w
j
are detected simultaneously for any user, an error in detection
is possible only if overlaps from the interfering users on wavelengths w
i
and w
j
are
bit synchronous (T), so the probability of error due to MAI for the proposed 2D
code families is
P

e
=
P
e
(N
w
i
) ×P
e
_
N
w
j
_
T
. (4.1)
Equation (4.1) is same as Eqn. (3.5) and is reproduced again for easier under-
standing. The wavelength allotment by the proposed construction is not uniform
when the number of wavelengths is odd, due to which the number of interfering
wavelengths (N
w
i
, N
w
j
) for the two wavelengths assigned to a user differ. For each
user, the probability of error due to MAI is calculated and the mean value of all
users is taken. To obtain curves corresponding to different number of active users,
the probability of error is determined by successively increasing the number of
rows of the generated array. The probability of error for the 2D code families
constructed using CRWOP for weight K

= 4 is found to be below 4 × 10
−4
for
upto 32 wavelengths as shown in Fig. 4.4.
Figure 4.5 shows a comparative view of the probability of error due to MAI
81
OCDMA Code Families based on a Novel CRWOP algorithm
0 50 100 150 200 250 300
10
−8
10
−7
10
−6
10
−5
10
−4
10
−3
Number of Active Users
P
r
o
b
a
b
i
l
i
t
y

o
f

E
r
r
o
r
W=7, T=9, K’=4, N
max
=12
W=12, T=13, K’=4, N
max
=36
W=21, T=23, K’=4, N
max
=110
W=32, T=33, K’=4, N
max
=256
Figure 4.4: Probability of error due
to MAI of 2D CRWOP-based code
families
50 100 150 200 250
10
−10
10
−8
10
−6
10
−4
10
−2
Number of active users
P
r
o
b
a
b
i
l
i
t
y

o
f

e
r
r
o
r
MWOOC: C
d
=1991, K’=10, N
max
=242
MWOOC: C
d
=2915, K’=12, N
max
=242
CRWOP: C
d
=1056, K’=4, N
max
=256
RWOP: C
d
=1128, K’=4, N
max
=248
Figure 4.5: Probability of error due
to MAI of 2D CRWOP-based, RWOP-
based and MWOOC families
among CRWOP-based, RWOP-based and MWOOC 2D code families. The com-
parison is based on equivalent cardinality of the 2D code families having varying
code dimensions. The probability of error due to MAI of the CRWOP-based 2D
code families is equivalent to that of the RWOP-based 2D code families. When
compared with the MWOOC 2D code families, the probability of error due to MAI
of the CRWOP-based 2D code families is higher at low number of active users and
lower when all users are interfering. At full cardinality, the probability of error due
to MAI of the CRWOP-based 2D code families is lower by more than one order
when compared with weight 10 MWOOC 2D code families at nearly half (0.53)
the code dimension of MWOOCs. The probability of error due to MAI of the
CRWOP-based 2D code families is marginally higher when compared with weight
12 MWOOC 2D code families at nearly one-third (0.36) the code dimension of
MWOOCs when all users are interfering.
An alternative comparative view of the probability of error due to MAI (based
on code dimension) of the 2D code families is analyzed. Figure 4.6 shows the
comparison of probability of error among reported (a) MWOOC [65], (b) RWOP-
based (Sec. 3.5.2) and (c) CRWOP-based 2D code families. Comparing between
(a) and (c), the probability of error of (c) is lower by a factor of 10
−2
at a spectral
82
4.4 CRWOP-based 2D OCDMA Code Families
5 10 15 20 25 30 35 40 45 50
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
Number of active users
P
r
o
b
a
b
i
l
i
t
y

o
f

e
r
r
o
r
(a)MWOOC
(b)RWOP
(c)CRWOP
Figure 4.6: Comparison of (a) MWOOC(5 ×41, 5, 1, 1); N
max
= 50; C
d
= 205, (b)
RWOP(10 × 19, 4, 2, 1); N
max
= 39; C
d
= 190 and (c) CRWOP(13 × 15, 4, 2, 1);
N
max
= 42; C
d
= 195 2D code families
efficiency of 0.2333 for CRWOP compared to 0.2439 for MWOOC. Comparison
between (b) and (c) shows that the probability of error is almost same at a spectral
efficiency of 0.2333 for CRWOP compared to 0.2053 for RWOP.
Table 4.8: Example 2D CRWOP-based code family for W = 5, T = 7 & K

= 4
Users W : T
U
1
(w
1
: t
1
t
7
), (w
4
: t
1
t
7
)
U
2
(w
3
: t
1
t
7
), (w
5
: t
1
t
7
)
U
3
(w
2
: t
2
t
6
), (w
4
: t
2
t
6
)
U
4
(w
1
: t
2
t
6
), (w
5
: t
2
t
6
)
U
5
(w
3
: t
3
t
5
), (w
4
: t
3
t
5
)
U
6
(w
2
: t
3
t
5
), (w
5
: t
3
t
5
)
Table 4.8 shows an example CRWOP-based 2D code family in the same rep-
resentation as the RWOP-based 2D code family in Table 3.7.
The first comparison (Comp1 in Table 4.9) considers the smallest possible
RWOP-based 2D code family with that of a CRWOP-based code family having
83
OCDMA Code Families based on a Novel CRWOP algorithm
Table 4.9: Comparison of RWOP-based and CRWOP-based 2D code families
Code FamilyCode DimensionCode WeightCardinality P
e
(W ×T) (K

) (N
max
) (MAI)
Comp1
RWOP 4 ×7 4 6 5.95 ×10
−5
CRWOP 5 ×7 4 6 0
Comp2
RWOP 6 ×11 4 13 1.03 ×10
−4
CRWOP 7 ×9 4 12 5.08 ×10
−5
Comp3
RWOP 8 ×15 4 26 2.10 ×10
−4
CRWOP 10 ×11 4 25 2.24 ×10
−4
Comp4
RWOP 14 ×27 4 79 1.76 ×10
−4
CRWOP 18 ×19 4 81 3.17 ×10
−4
the same cardinality. Other comparisons (Comp2, Comp3 and Comp4) are consid-
ered for equivalent higher cardinality to point out the advantages of the CRWOP
algorithm. In Comp1, the CRWOP-based code family uses an extra wavelength to
accomodate the same number of users resulting in lower probability of error due to
MAI. It should be noted from the other comparisons that the code dimension de-
creases and the probability of error due to MAI increases with increasing cardinal-
ity of the CRWOP-based code families. The advantage of the CRWOP-based code
families is the feasibility of large cardinality code families with reduced temporal
spreading. Lower temporal spreading allows for the feasibility of integrated-optic
devices to generate OCDMA codes.
4.5 CRWOP-based 3D OCDMA Code Families
The CRWOP algorithm is used to construct 3D space-wavelength-time code fam-
ilies. The construction of K
′′
= 8 3D code families with equal number of wave-
lengths and space channels (S = W) and with number of wavelengths more than
space channels (S < W) is explained in 4.5.1. As in the case of the CRWOP-based
84
4.5 CRWOP-based 3D OCDMA Code Families
2D construction, this construction also gives an improvement in cardinality for
equivalent code dimension in comparison with the RWOP-based 3D code families.
The performance analysis in terms of cardinality, spectral efficiency, probability
of error due to MAI and comparison of the constructed 3D code families with
the RWOP-based 3D code families (Sec. 3.6.2) and other previously reported
literature are shown in 4.5.2.
4.5.1 Construction of CRWOP-based 3D Code Families
The construction of these 3D code families (S = W and S < W) is similar to
that of the RWOP-based 3D code families (Sec. 3.6.1) and is explained briefly.
The CRWOP algorithm is used for both wavelength as well as spatial allocation.
Each pair of fibers is paired with all the wavelength pairs of the corresponding
row. As in the case of 2D, each row is assigned a distinct 1D OOC spread in the
time domain. This construction also ensures a maximum crosscorrelation of ‘1’
between any two users and a maximum out of phase autocorrelation of ‘4’. The
weight of these constructed 2D code families is given by K
′′
= 4K. The factor ‘4’
arises due to the simultaneous transmission of the 1D OOC along four channels (‘2’
fibers, each using ‘2’ wavelengths per user). In these constructions, the required
number of 1D OOCs is given by n = ⌈W/2⌉. So the optimum temporal length of
these 2D code families is given by T
opt
= K(K − 1)n + 1, where K is the weight
of the employed 1D OOC family. An example 3D code family constructed using
S = W = 7 is shown in Table 4.10. The other parameters in the example for
S = W = 7 are n = 4, K = 2, K
′′
= 8 and T = T
opt
= 9. The code family shown
in the example can be represented as (7 ×7 ×9, 8, 4, 1), with N
max
= 36.
An example for constructing 3D code families with S < W where S = 4 and
W = 5 is shown in Table 4.11. This construction is also similar to that of the
S < W 3D code families using the RWOP algorithm shown in Sec. 3.6.1. This
85
OCDMA Code Families based on a Novel CRWOP algorithm
Table 4.10: Example of 3D code construction (S = W = 7, K
′′
= 8)
Fiber Allocation Wavelength Allocation 1D OOC
(s
1
, s
5
) (s
4
, s
6
) (s
3
, s
7
) (w
1
, w
5
) (w
4
, w
6
) (w
3
, w
7
) C
1
= [1, 9]
(s
2
, s
5
) (s
1
, s
6
) (s
4
, s
7
) (w
2
, w
5
) (w
1
, w
6
) (w
4
, w
7
) C
2
= [2, 8]
(s
3
, s
5
) (s
2
, s
6
) (s
1
, s
7
) (w
3
, w
5
) (w
2
, w
6
) (w
1
, w
7
) C
3
= [3, 7]
(s
4
, s
5
) (s
3
, s
6
) (s
2
, s
7
) (w
4
, w
5
) (w
3
, w
6
) (w
2
, w
7
) C
4
= [4, 6]
code family can be represented as (4 × 5 × 7, 8, 4, 1), with N
max
= 12. Codes of
each user (U
1
, U
2
, . . . , U
12
) in the (4 × 5 × 7, 8, 4, 1) 3D code family are shown in
Table 4.12.
Table 4.11: Example of S < W 3D code construction (S = 4, W = 5, K
′′
= 8)
Fiber Allocation Wavelength Allocation 1D OOC
(s
1
, s
3
) (s
2
, s
4
) (w
1
, w
4
) (w
3
, w
5
) C
1
= [1, 7]
(s
2
, s
3
) (s
1
, s
4
) (w
2
, w
4
) (w
1
, w
5
) C
2
= [2, 6]
(s
1
, s
3
) (s
2
, s
4
) (w
3
, w
4
) (w
2
, w
5
) C
3
= [3, 5]
Alternatively, construction of the 3D S = W code families is also possible
by choosing the cardinality as the first parameter. To construct a 3D code family
with a cardinality greater than or equal to N
g
, the required number of wavelengths
(equal to the number of fibers) is W =
_
2
3
_
N
g
+ 1
_
.
4.5.2 Analysis of CRWOP-based 3D Code Families
The cardinality of the S = W and S < W 3D code families is N
max
= n(W −n)
2
and N
max
= n(W − n)(S − n

) respectively, where n = ⌈W/2⌉ and n

= ⌈S/2⌉.
Figure 4.7 shows the cardinality of CRWOP-based, RWOP-based and SPP 3D
code families for varying code dimensions. For equivalent code dimension, the
cardinality of the CRWOP-based 3D code families is marginally higher than that
of the RWOP-based 3D code families. The cardinality of the SPP 3D code families
is shown for large weights (19 and 53) leading to low cardinality, since these same
weights were considered for the comparison of probability of error due to MAI.
86
4.5 CRWOP-based 3D OCDMA Code Families
Table 4.12: Codes of all users for the same example as Table 4.11
U
1
w
1
w
2
w
3
w
4
w
5
s
1
C
1
C
1
s
2
s
3
C
1
C
1
s
4
U
2
w
1
w
2
w
3
w
4
w
5
s
1
C
1
C
1
s
2
s
3
C
1
C
1
s
4
U
3
w
1
w
2
w
3
w
4
w
5
s
1
s
2
C
1
C
1
s
3
s
4
C
1
C
1
U
4
w
1
w
2
w
3
w
4
w
5
s
1
s
2
C
1
C
1
s
3
s
4
C
1
C
1
U
5
w
1
w
2
w
3
w
4
w
5
s
1
s
2
C
2
C
2
s
3
C
2
C
2
s
4
U
6
w
1
w
2
w
3
w
4
w
5
s
1
s
2
C
2
C
2
s
3
C
2
C
2
s
4
U
7
w
1
w
2
w
3
w
4
w
5
s
1
C
2
C
2
s
2
s
3
s
4
C
2
C
2
U
8
w
1
w
2
w
3
w
4
w
5
s
1
C
2
C
2
s
2
s
3
s
4
C
2
C
2
U
9
w
1
w
2
w
3
w
4
w
5
s
1
C
3
C
3
s
2
s
3
C
3
C
3
s
4
U
10
w
1
w
2
w
3
w
4
w
5
s
1
C
3
C
3
s
2
s
3
C
3
C
3
s
4
U
11
w
1
w
2
w
3
w
4
w
5
s
1
s
2
C
3
C
3
s
3
s
4
C
3
C
3
U
12
w
1
w
2
w
3
w
4
w
5
s
1
s
2
C
3
C
3
s
3
s
4
C
3
C
3
For S = W and S < W 3D code families, the spectral efficiency is η(3D) =
n(W−n)
2
/(W
2
(K(K−1)n+1)) and η(3D) = n(W−n)(S−n

)/(SW(K(K−1)n+
1)) respectively. As shown in Fig. 4.8, the spectral efficiency of the S = W 3D
code families varies from 0.08 to 0.12 and the spectral efficiency of the S < W 3D
code families varies from 0.1 to 0.17 for the given range of code dimension. The
87
OCDMA Code Families based on a Novel CRWOP algorithm
0 1000 2000 3000 4000 5000 6000 7000 8000
0
100
200
300
400
500
600
700
800
900
C
a
r
d
i
n
a
l
i
t
y
Code Dimension
RWOP−3D (S=W): K’’=8
RWOP−3D (S<W): K’’=8
CRWOP−3D (S=W): K’’=8
CRWOP−3D (S<W): K’’=8
SPP−3D (S>W): K’’=S=19
SPP−3D (S>W): K’’=S=53
Figure 4.7: Cardinality of CRWOP-
based, RWOP-based and SPP 3D code
families
0 1000 2000 3000 4000 5000 6000 7000 8000
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
S
p
e
c
t
r
a
l

E
f
f
i
c
i
e
n
c
y
Code Dimension
RWOP: S=W, K’’=8
CRWOP: S=W, K’’=8
RWOP: S<W, K’’=8
CRWOP: S<W, K’’=8
Figure 4.8: Spectral efficiency of
CRWOP-based and RWOP-based 3D
code families
spectral efficiency of the CRWOP-based 3D code families is marginally higher
than that of the RWOP-based 3D code families. The spectral efficiency of the
SPP 3D code families 1/S which is very low for the large weights considered in
our comparisons.
The probability of error due to MAI corresponding to channels (s
i
, w
i
), (s
i
, w
j
), (s
j
, w
i
)
and (s
j
, w
j
) are represented as P
e
(N
s
i
,w
i
), P
e
_
N
s
i
,w
j
_
, P
e
_
N
s
j
,w
i
_
and P
e
_
N
s
j
,w
j
_
.
Since the four 1D OOCs are detected simultaneously for any user, an error in
detection is possible only if overlaps from the interfering users on all four channels
are bit synchronous (T), so the probability of error due to MAI for the proposed
3D code families is
P
′′
e
=
P
e
(N
s
i
,w
i
) P
e
_
N
s
i
,w
j
_
P
e
_
N
s
j
,w
i
_
P
e
_
N
s
j
,w
j
_
T
3
. (4.2)
The probability of error for the S = W and S < W 3D code families con-
structed using CRWOP for weight K

= 8 is found to be below 4 ×10
−8
as shown
in Fig. 4.9. The probability of error of the S < W 3D code families is almost
same as that of the S = W 3D code families for equivalent code dimension.
88
4.5 CRWOP-based 3D OCDMA Code Families
0 100 200 300 400 500 600 700 800
10
−18
10
−16
10
−14
10
−12
10
−10
10
−8
10
−6
Number of Active Users
P
r
o
b
a
b
i
l
i
t
y

o
f

E
r
r
o
r
S=W=6, T=7, K’’=8, N
max
=27
S=W=12, T=13, K’’=8, N
max
=216
S=W=18, T=19, K’’=8, N
max
=729
S=4, W=7, T=9, K’’=8, N
max
=24
S=7, W=16, T=17, K’’=8, N
max
=192
S=6, W=31, T=33, K’’=8, N
max
=720
Figure 4.9: Probability of error due to
MAI of CRWOP-based S = W and
S < W 3D code families
20 30 40 50 60 70 80 90 100 110
10
−14
10
−13
10
−12
10
−11
10
−10
10
−9
10
−8
10
−7
Number of active users
P
r
o
b
a
b
i
l
i
t
y

o
f

e
r
r
o
r
CRWOP: S=4, W=15, T=17, N
max
=112
RWOP: S=W=8, T=15, N
max
=98
SPP: S=53, W=6, T=17, N
max
=102
Figure 4.10: Probability of error due to
MAI of CRWOP-based, RWOP-based
and SPP 3D code families
The comparison of probability of error due to MAI between CRWOP-based,
RWOP-based and SPP 3D code families is based on highest probability of error
due to MAI. Figure 4.10 shows the comparison of probability of error among
CRWOP-based, RWOP-based and SPP 3D code families for probability of error
less than 2 × 10
−8
. At high number of active users, the probability of error is
equivalent for all the 3D code families with a spectral efficiency of 0.11 for the
CRWOP-based 3D code families, 0.102 for the RWOP-based 3D code families and
0.019 for the SPP 3D code families. At low number of active users, the probability
of error of the SPP 3D code families is lower than that of the CRWOP-based and
RWOP-based 3D code families.
Based on code dimension, comparison of the probability of error due to MAI of
(a) SPP [102], (b) RWOP-based and (c) CRWOP-based 3D code families is shown
in Fig. 4.11. Analyzing the characteristics between (a) and (c), the probability
of error of (c) is lower by a factor of 10
−6
at a spectral efficiency of 0.1154 for
CRWOP compared to 0.1423 for SPP. From the characteristics of (b) and (c), the
probability of error is almost same at a spectral efficiency of 0.1154 for CRWOP
compared to 0.09 for RWOP.
89
OCDMA Code Families based on a Novel CRWOP algorithm
50 100 150 200 250
10
−14
10
−12
10
−10
10
−8
10
−6
10
−4
10
−2
Number of active users
P
r
o
b
a
b
i
l
i
t
y

o
f

e
r
r
o
r
(a)SPP
(b)RWOP
(c)CRWOP
Figure 4.11: Comparison of (a) SPP(7 × 5 × 53, 7, 0, 1); N
max
= 264; C
d
= 1855,
(b) RWOP(10 × 10 × 19, 8, 4, 1); N
max
= 171; C
d
= 1900 and (c) CRWOP(12 ×
12 ×13, 8, 4, 1); N
max
= 216; C
d
= 1872 3D code families
4.6 Summary
The proposed CRWOP-based code families give better performance than the
RWOP-based code families discussed in Chapter 3 and also better than those pub-
lished earlier. The code families are suitable for networks which are to be deployed
with low error probabilities. Due to the completeness of the wavelength/space al-
location algorithm, the cardinality and spectral efficiency of the CRWOP-based
code families are marginally higher than that of the RWOP-based code families.
The code dimension of the CRWOP-based 2D code families is lower than that of
the RWOP-based 2D code families leading to higher spectral efficiency at equiva-
lent cardinality. However, the probability of error due to MAI is marginally higher.
The probability of error of the CRWOP-based and RWOP-based code families is
equivalent to that of MWOOCs and SPP codes for full cardinality. The probability
of error however is higher than SPP codes for small number of active users while
yielding better spectral efficiency. The CRWOP-based 2D and 3D code families
90
4.6 Summary
have lower probability of error at lower spectral efficiency when compared with
MWOOCs and SPP codes for equivalent code dimension.
A comparison based on code dimension shows that the CRWOP-based 2D and
3D code families have lower probability of error at higher spectral efficiency as
compared to the RWOP-based code families. A close look at the comparison of
RWOP and CRWOP apparently indicates that the probability of error differs by
a small margin. However, the cardinality supported by CRWOP with 2 additional
fiber paths is higher by a factor 216/171. This trend has been supported by Fig.
4.11 for 3D case which in turn becomes 42/39 for the 2D case as shown in Fig.
4.6.
91
C H A P T E R 5
Miniaturization of OCDMA Code
Generation
5.1 Introduction
Based on some of the OCDMA code families in chapters 2, 3 and 4, designs for
the generation of 1D, proposed 2D multipulse per row (MPR) and 3D multipulse
per plane (MPP) code families using compact integrated-optics are considered.
Generation of these codes in the picosecond regime using optical fiber delay lines
is difficult due to the micron precision fiber lengths required. Optical delay of
the order of picoseconds and sub-picoseconds is easier to generate by integrated-
optics. The integrated optic devices discussed here are based on Titanium in-
diffused Lithium Niobate (Ti:LiNbO
3
) technology. The basic building blocks of
these deivces are directional couplers which have a wide range of applications in
optical communications. Directional couplers are more commonly used as 3dB
couplers to split the intensity of the light at the input port equally to the two
Miniaturization of OCDMA Code Generation
output ports. In order to design compact devices, the use of Zero-Gap Directional
Coupler (ZDC) [122] is explored. The ZDC is also referred to as a Two Mode
Interference Coupler (TMI) [122, 123]. The ZDCs are designed as TE-TM mode
splitters to be able to use the birefringence property of LiNbO
3
. The birefringence
property of a material causes an incident ray of light to be divided into an ordi-
nary ray and an extra-ordinary ray [124]. The ordinary ray obeys Snell’s law, but
the extra-ordinary ray does not obey Snell’s law [125]. The ordinary and extra-
ordinary rays encounter different refractive indices which arise due to anisotropy
in the crystal structure [124]. Hence, the two different refractive indices are known
as ordinary refractive index (n
o
) and extra-ordinary refractive index (n
e
).
A TE-TM mode splitter is a device that splits input unpolarised light into two
different polarisations: transverse electric (TE) and transverse magnetic (TM).
TE-TM mode splitters in LiNbO
3
have been reported by proton exchange on one
arm of the Y-junction to guide the TM mode and Ti diffusion in the other arm to
guide the TE mode [126]. Another configuration uses Ti diffused input arm, Ni
diffused branched output channel for TM mode, and MgO induced LiNbO
3
out-
diffusion in the straight output channel for TE mode [127]. In another branched
channel configuration using codiffused Zn and Ni, the splitting of TE and TM
modes is obtained by using a lower diffusion temperature for the branched output
channel to guide the TM mode [128]. More recent developments include integra-
tion of a photonic crystal polarization beam splitter with a waveguide bend, the
performance of the individual components and the performance of the integrated
device [129] and a silicon photonic circuit wavelength filter using polarization split-
ters, polarization rotators and a ring resonator [130].
The TE mode has no electric field in the direction of propagation and the
TM mode has no magnetic field in the direction of propagation. In case of z-
cut LiNbO
3
, the TE mode encounters the ordinary refractive index and the TM
94
5.2 Zero-Gap Directional Coupler
mode encounters the extra-ordinary refractive index [131]. In Ti:LiNbO
3
, the
ordinary refractive index is approximately 2.22 and the extra-ordinary refractive
index is approximately 2.15 at a wavelength of 1.3µm. The simulation of the
refractive index profiles of Ti:LiNbO
3
waveguides and directional couplers [119],
the propagation constants of directional couplers and the crtical coupling lengths
of the TE and TM modes [120] is also discussed.
In this chapter, design of a ZDC as a TE-TM mode splitter is proposed. The
application of the TE-TM mode splitter in designing 1D, 2D and 3D OCDMA
code generators is worked out. The designed miniature OCDMA code generators
are compared with similar devices replacing the ZDC with a 3dB power splitting
Y-junction. However, details of designs are not presented in this thesis as such
work would be beyond the scope of the thesis which is addressing code construction
design.
5.2 Zero-Gap Directional Coupler
Conventional integrated optic directional couplers have a gap between the two
waveguides in the coupling region as shown in Fig. 5.1. Zero-gap directional
couplers reduce the gap between the waveguides to zero resulting in a coupling
region which has twice the width of the individual waveguides as shown in Fig.
5.2. The coupling region of the ZDC supports two modes: a symmetric and an
asymmetric mode. Separation of TE and TM modes from one another needs to
be explored for the application to pulse pattern generation.
The proposed design is based on Ti:LiNbO
3
technology, and therefore, involves
computing of the ordinary and extra-ordinary refractive indices for a set of pa-
rameters as given in [119]. Effective-Index based Matrix Method (EIMM) has
been used in [120] to compute the propagation constants and hence the critical
95
Miniaturization of OCDMA Code Generation
L
c
L
c
=coupling length w=width, g=gap,
B D
w w θ
A C
i/p w w
g
Figure 5.1: A conventional inte-
grated optic directional coupler
L
c
l
1
l
2
l
3
l
4
TM
TE
A
B
C
D
i/p w
w
w
w θ
2w
Figure 5.2: Proposed integrated optic
zero-gap directional coupler
coupling lengths of the ZDC for TE and TM polarizations. Based on the criti-
cal coupling lengths derived from the above cited procedures, normalized power
coupling equation is used to choose a coupling length as required by the desired
application.
The following subsections discuss the simulation parameters, plots of ordinary
and extra-ordinary refractive indices, propagation constant versus excitation effi-
ciency of TE and TM modes for a simulated set of parameters. From the simulated
results, a coupling length is chosen for the desired TE-TM splitting and a mask
design for fabrication of ZDC is shown.
5.2.1 Determination of Critical Coupling Lengths
The refractive index profile is modelled here as the sum of the refractive index
given by the Sellmeier equations and the change in refractive index induced by
indiffused Titanium. The Sellmeier equations governing the ordinary and extra-
ordinary refractive indices are given by
n
2
o
= 4.9048 −[0.11768/(0.04750 −w
2
)] −0.027169w
2
, (5.1a)
n
2
e
= 4.5820 −[0.099169/(0.04432 −w
2
)] −0.021950w
2
, (5.1b)
where w is the wavelength of operation in micrometers. The change in ordinary
refractive index (∆n
o
) and the change in extra-ordinary refractive index (∆n
e
)
[119] are given by
96
5.2 Zero-Gap Directional Coupler
∆n
o,e
(x, z, w) = A
o,e
(w, C
s
)[C(x, z)]
αo,e
, (5.2)
where A
o,e
, α
o,e
are constants independent of x− and z− coordinates, C
s
is the
surface concentration of indiffused Titanium and C(x, z) is the concentration of
indiffused Titanium. The concentration of indiffused Titanium for coupled waveg-
uides [119] as shown in Fig. 5.3 is represented by
LiNbO
3
Ti
x
z
y
0
Ti
S
w
w
τ
Figure 5.3: Deposited Titanium strips on a LiNbO
3
substrate for a coupler
C(x, z, t) = 0.25C
0
[erf(τ +z)/d
z
+ erf(τ −z)/d
z
]
×[erf(w +g/2 +x)/d
x
+ erf−(x +g/2)/d
x
+erf(w +g/2 −x)/d
x
+ erf(x −g/2)/d
x
] (5.3)
where w is the width of the single-mode waveguide, g is the gap between the
waveguides, τ is the thickness of the Titanium layer to be deposited, t is the
diffusion time and C
0
is the initial concentration of Titanium in LiNbO
3
at t = 0
for 0 ≤ z ≤ τ. Another parameter used in the simulation is diffusion temperature
(Υ), which we have considered to be 1050

C in all our simulations.
In our model, g = 0 is considered because of the fact that ZDC implies the
97
Miniaturization of OCDMA Code Generation
mutual joining of the two Titanium strips. However, the result for g = 0 as a
wavelength division multiplexer/demultiplexer is included in [122]. The parame-
ters used for simulating the ZDC in [122] are w = 7µm, 2w = 14µm, τ = 0.095µm,
t = 6 hours, w = 1.318µm and Υ = 1050

C. The operating wavelength of 1.3µm
is chosen corresponding to the second window in optical fiber communications.
Refractive index variations in the considered wavelength (w = 1.3µm) is required
as a ready reference for the ZDC design. For a ZDC shown in Fig. 5.2, w, τ and
t should be varied and checked such that a single mode is excited at the required
wavelength of operation. Replacing w with 2w, the coupling region must excite
two modes. If the coupling region does not excite two modes, w, τ and t have to be
varied repeatedly until two modes are excited in the coupling region. Simulation
of refractive index profiles for w = 6.3µm, 2w = 12.6µm, τ = 0.095µm, t = 6
hours and w = 1.3µm is carried out. The ordinary and extra-ordinary refractive
index profiles are shown in Figures 5.4 and 5.5 respectively.
−15 −10 −5 0 5 10 15
2.22
2.221
2.222
2.223
2.224
2.225
2.226
2.227
2.228
2.229
X−axis of substrate
O
r
d
i
n
a
r
y

r
e
f
r
a
c
t
i
v
e

i
n
d
e
x
Single Mode: Width = 6.3 microns
Double Mode: Width = 12.6 microns
Figure 5.4: Ordinary refractive in-
dex profiles of single-mode and double-
mode regions
−15 −10 −5 0 5 10 15
2.144
2.146
2.148
2.15
2.152
2.154
2.156
2.158
2.16
X−axis of substrate
E
x
t
r
a

o
r
d
i
n
a
r
y

r
e
f
r
a
c
t
i
v
e

i
n
d
e
x
Single Mode: Width = 6.3 microns
Double Mode: Width = 12.6 microns
Figure 5.5: Extra-ordinary refrac-
tive index profiles of single-mode and
double-mode regions
The obtained refractive index profiles along the X-direction are sectored into
3000 layers which are approximated to staircase-type step-index profiles according
to [120]. The propagation of TE and TM modes through the ZDC is governed by
98
5.2 Zero-Gap Directional Coupler
the excitation efficiency given by
η =
¸
¸
¸
¸
E
+
g2
E
+
1
¸
¸
¸
¸
2
=
¸
¸
¸
¸
s

22
s
11
−s
21
s

12
s
11
(s

11
s

22
−s

21
s

12
)
¸
¸
¸
¸
2
, (5.4)
where E
+
1
is the incident electric field on the first layer, E
+
g2
is the incident electric
field on an intermediate layer g2 having highest refractive index, the s-parameters
are elements of a transmission matrix corresponding to all 3000 layers and the
s

-parameters are elements of a transmission matrix from the first layer to the
g2
th
layer [120]. Each value of θ
1
yields corresponding values of β and η, where
θ
1
is the angle of incidence of the electric field on the first layer and β is the
propagation constant of the electromagnetic wave through all the layers. The
relationship between β and θ
1
is given by
β =

w
n
1
sin θ
1
, (5.5)
where w is the operating wavelength and n
1
is the refractive index of the first layer.
The propagation constant versus excitation efficiency of TE and TM modes for the
ZDC is shown in Figures 5.6 and 5.7 respectively. The values of β corresponding to
the peaks in Figures 5.6 and 5.7 represent the propagation constants of a symmetric
mode (β
s
) and an asymmetric mode (β
a
). The critical coupling length (L
c
) for
100% power transfer [124] is given by
L
c
=
π
β
s
−β
a
. (5.6)
From the propagation constants obtained from Figures 5.6 and 5.7 and using
Eqn. (5.6), the critical coupling lengths of TE (L
c
(TE)) and TM (L
c
(TM)) modes
are found to be 589µm and 286µm respectively. Power transfer equations [124]
for a directional coupler are given by
99
Miniaturization of OCDMA Code Generation
10.732 10.734 10.736 10.738 10.74 10.742 10.744 10.746
0
5
10
15
20
25
30
35
Propagation Constant
E
x
c
i
t
a
t
i
o
n

E
f
f
i
c
i
e
n
c
y
TE modes, λ=1.3 microns
Figure 5.6: Propagation constant ver-
sus excitation efficiency of TE modes
10.38 10.385 10.39 10.395 10.4 10.405 10.41
0
0.05
0.1
0.15
0.2
0.25
Propagation Constant
E
x
c
i
t
a
t
i
o
n

E
f
f
i
c
i
e
n
c
y
TM modes, λ = 1.3 microns
Figure 5.7: Propagation constant ver-
sus excitation efficiency of TM modes
P
C
= 1 −sin
2
β
c
y,
P
D
= sin
2
β
c
y, (5.7)
where P
C
is the normalized power at port C, P
D
is the normalized power at port
D, β
c
=
π
2Lc
and y is the direction of propagation.
5.2.2 Design of ZDC
The objective here is to observe the variations in crosstalk between TE and TM
modes at the output ports with respect to the coupling length of the ZDC. The
crosstalk is calculated as follows:
100
5.2 Zero-Gap Directional Coupler
crosstalk(TE) =
power of TM mode at D
power of TE mode at D
crosstalk(TM) =
power of TE mode at C
power of TM mode at C
(5.8)
The propagation length for which crosstalk is minimum for both TE and TM
modes is chosen as the coupling length of the ZDC. Figure 5.8 shows the crosstalk
levels for TE and TM modes for a range of coupling lengths corresponding to a
wavelength of 1.3µm. The best achievable crosstalk is found to be -29.2 dB and
-31.2 dB for TE and TM modes respectively at a coupling length of 577µm. The
total length of the splitter including the length of the input and output arms is
less than 15mm.
5.74 5.75 5.76 5.77 5.78 5.79 5.8
x 10
−4
−40
−35
−30
−25
Coupling length (meters)
C
r
o
s
s
t
a
l
k

(
d
B
)
Port D
Port C
Figure 5.8: Propagation length versus crosstalk
In the design, distance between the two input ports is chosen to be the same as
the distance between the two output ports, which is 125µm. This ensures proper
coupling of optical fibers with the device. A value of θ chosen as 1

is observed
to provide a trade-off between bending loss and electro-magnetic coupling at the
input and output junctions of the double mode region. It may be noted that, the
bending loss is low for θ = 0.6

, but the coupling of light is also higher at the
junctions, which in turn adds to the coupling length of the device. Coupling of
light from one waveguide to another is possible only when the distance between
101
Miniaturization of OCDMA Code Generation
the two waveguides is less than 9µm. Experiments show that, increasing θ from
0.6

to 1

, reduces the coupling at the junctions of the device to negligible levels.
The total length of the ZDC is the sum of lengths l
1
, l
2
, L
c
, l
3
and l
4
(shown in
Fig. 5.2), which comes to 11739.3µm.
5.3 Ti:LiNbO
3
based 1D OCDMA Code Gener-
ation
Temporal spread 1D OCDMA codes can be generated by using the TE-TM splitter
proposed in Section 5.2 as well as a 3dB power splitting Y-junction as shown in
subsections 5.3.1 and 5.3.2 respectively. The delay between pulses for the two
designs is analyzed by assuming a standard dimension of LiNbO
3
crystals to be
50mm×50mm×1mm. This dimension has normally been used as a standard in
various applications.
5.3.1 ZDC Based Code Generation
A miniaturized OCDMA code generator using a ZDC (MOCG:Z) on a standard
LiNbO
3
crystal is shown in Fig. 5.9. The code generator is designed by extending
and combining the output arms of the TE-TM splitter (Fig. 5.2). The code gener-
ator splits a single input pulse into two output pulses spread in the time domain.
The two output pulses of the code generator have orthogonal polarizations (TE
and TM). The issues in designing the 1D OCDMA code generator are discussed
and the calculations involved to find the time delay between the two pulses for
the 1D OCDMA code generator in an ideal case are shown below.
102
5.3 Ti:LiNbO
3
based 1D OCDMA Code Generation
L
c
i/p
A
B
C
D
w
w
2w
θ
TM
TE
w’
w’’
Figure 5.9: Miniature 1D OCDMA code generator using ZDC
Design Considerations:
The 1D OCDMA code generator shown in Fig. 5.9 has a length of 50mm and
a width of 10mm. The curved waveguide is designed to have negligible bending
loss by drawing three arcs with a bending radius of approximately 15mm. Start-
ing from the output junction of the ZDC, the length of the curved and straight
waveguides in Fig. 5.9 are 46861.6142µm and 42762.9455µm respectively. Figure
5.9 shows the maximum possible length of the curved waveguide on a standard
LiNbO
3
crystal for negligible bending loss.
The length of the curved waveguide can be increased further to completely span
over a standard LiNbO
3
crystal, but the bending radius has to be decreased from
15mm. Waveguide bending loss in Ti:LiNbO
3
increases with decreasing bending
radius.
Numerical Results:
Maximum delay between the two output pulses can be obtained if the curved
waveguide guides the TE mode and the straight waveguide guides the TM mode.
As shown in Fig. 5.2, for an input in port A, port C outputs TM mode and port
D outputs TE mode. So, the input to the miniature 1D OCDMA code generator
should be given in the lower input port (port B) resulting in TE mode at port C
and TM mode at port D.
The time taken for the TE mode (t
te
) to propagate through the curved waveg-
103
Miniaturization of OCDMA Code Generation
uide is the ratio between length of the curved waveguide (l

) to the velocity of
the TE mode (v
te
). Similarly, the time taken for the TM mode (t
tm
) to propa-
gate through the straight waveguide is the ratio between length of the straight
waveguide (l
′′
) to the velocity of the TM mode (v
tm
).
So the time delay between the two pulses is given by
∆t = t
te
−t
tm
. (5.9)
The velocities of the TE and TM modes are given by
v
te
=
c
n
o
and (5.10a)
v
tm
=
c
n
e
, (5.10b)
where c is the velocity of light, n
o
is the ordinary refractive index and n
e
is the
extra-ordinary refractive index. Hence, the time delay between the two pulses can
be written as
∆t =
l

n
o
−l
′′
n
e
c
. (5.11)
For l

= 46861.6142µm, l
′′
= 42762.9455µm (Sec. 5.3.1), n
o
= 2.226188 and
n
e
= 2.154467 (Sec. 5.2.1), the time delay between the two pulses, ∆t = 4063.8 ×
10
−14
. So the maximum possible delay between the two output pulses on a LiNbO
3
crystal of dimensions 50mm× 50mm × 1mm is 40.638 picoseconds. Relating to
OCDMA, all the pulses of a code family should be contained within the time frame
of this maximum delay.
In OCDMA, the pulse-width of the optical pulses is the chip time (T
c
) and the
number of chips in a code family is the temporal length (T - 2.3.3). If T
c
= 1ps,
then T = ∆t/T
c
= 40 chips can be accommodated in 40ps. The temporal length
104
5.3 Ti:LiNbO
3
based 1D OCDMA Code Generation
of a 1D OCDMA code family is given by
T ≥ K(K −1)N
max
+ 1, (5.12)
where K is the weight of the code family and N
max
is the cardinality or maximum
number of users in the code family. Since the output of the miniature 1D OCDMA
code generator is only two pulses, the weight of the code families generated is
K = 2. Substituting for K and rearranging Eqn. (5.12), the cardinality of a 1D
code family can be given by
N
max

T −1
2
. (5.13)
Since T = 40, N
max
= 19 under ideal conditions without taking the dispersion
effects into account. Increasing the value of T
c
, decreases T and the cardinality of
1D code family would be low. The cardinality of 1D code family can be increased
by decreasing the value of T
c
(using narrow pulse-widths). For generating contin-
uous data stream, the pulse repetition rate of the pulsed source should be 25GHz
(which is equal to
1
40ps
).
Different polarization states of the input pulse to the miniaturized OCDMA
code generator are considered and the output is analyzed. If the input pulse is
randomly polarized, the output would be a TE mode and a TM mode of unequal
amplitudes as shown in Fig 5.10. The amplitudes of the TE and TM modes vary
from zero to maximum depending on the polarization of the input pulse. For an
input pulse polarized linearly at 45

, the two output pulses would be of equal
amplitude. Hence a polarizer should be used before the MOCG:Z, as shown in
Fig. 5.11, to obtain pulses of equal amplitude as is necessary in OCDMA.
105
Miniaturization of OCDMA Code Generation
Random
Polarization
Pulsed Laser
Source
t
MOCG
TM TE
Figure 5.10: Input and output pulses of MOCG:Z
t
TM TE
Polarizer
MOCG
Polarization
Random
Source
Pulsed Laser
Polarization
45 Linear
o
Figure 5.11: Input and output pulses using a polarizer before the MOCG:Z
5.3.2 Y-junction Based Generation
The use of a 3dB power splitting Y-junction in place of the TE-TM mode splitter to
generate 1D OCDMA codes is shown in Fig. 5.12. The intensity of an input optical
pulse to a miniaturized OCDMA code generator using Y-junction (MOCG:Y)
would be split equally to the two output arms. As in the case of MOCG:Z,
the output of MOCG:Y also consists of two pulses spread in the time domain.
Further analysis of delay between the two output pulses corresponding to different
polarization states of the input pulse considers the same values of l

and l
′′
. If the
input pulse is TE polarized, both the split pulses would be TE polarized and the
delay between them at the output given in Eqn. (5.11) would change to
∆t =
(l

−l
′′
)n
o
c
. (5.14)
For the same values of n
o
, l

and l
′′
, ∆t = 3041.4 × 10
−14
which is 30.4 pi-
coseconds. For a chip time (T
c
) of 1ps, 30 chips can be accommodated in 30ps
leading to a cardinality of 14 as compared to 19 when a TE-TM mode splitter is
used instead of a 3dB power splitting Y-junction. In this case, the pulse repetition
106
5.3 Ti:LiNbO
3
based 1D OCDMA Code Generation
i/p
w’
w’’
w
θ
Figure 5.12: Miniaturized 1D OCDMA code generator using a 3dB power splitting
Y-junction
rate of the pulsed source should be 1/30ps = 33GHz. Such repetition rates are
achievable by using a continuous wave laser along with an external modulator.
If the input pulse is TM polarized, both the split pulses would be TM polarized
and the delay between them at the output given in Eqn. (5.11) would change to
∆t =
(l

−l
′′
)n
e
c
. (5.15)
For the same values of n
e
, l

and l
′′
, ∆t = 2943.4 ×10
−14
which is 29.4 picosec-
onds. For a chip time (T
c
) of 1ps, 29 chips can be accommodated in 29ps leading
to a cardinality of 14. It can be noted that, even though excitation of a single po-
larization and its power splitting leads to lower cardinality, the delay between the
two output pulses would not change due to polarization mode dispersion (PMD).
Hence, compensation for difference in delay between the two pulses due to PMD
is not needed in such cases.
For an input pulse polarized linearly at 45

, equal intensities of both the TE
and TM modes would be split and propagate through l

and l
′′
. The time taken
for the TE and TM modes to propagate a distance of l

is 347.74ps and 336.53ps
respectively. The time taken for the TE and TM modes to propagate a distance
of l
′′
is 317.32ps and 307.10ps respectively. Hence, a single 1ps input pulse gives
four 1ps pulses spread in the time domain at the output.
An advantage of using the 3dB power splitting Y-junction is its shorter length
107
Miniaturization of OCDMA Code Generation
compared to the TE-TM splitter. The Y-junction would be shorter by about 4mm
owing to the reduction due to the abscence of coupling length (L
c
in Fig. 5.2) and
bent waveguides (l
2
in Fig. 5.2).
5.3.3 Delay Comparison
A comparison of the delay between the two output pulses (∆t) for different con-
figurations is shown in Fig. 5.13 as a function of differential length (∆l). The
differential length is given by ∆l = (l

−l
′′
), where the value of l
′′
is constant and
the value of l

is decreased from its maximum of 46861.6142µm to a minimum
of 42762.9455µm(= l
′′
). Curve (a) shows a linear increase in ∆t with increasing
∆l for an input pulse at port B of Fig. 5.9. Even for equal lengths of l

and l
′′
(∆l = 0), a delay between the two output pulses exists due to the difference in n
o
and n
e
.
In case of curve (b), the delay is same as that of curve (a) for ∆l = 0 and the
delay decreases linearly till zero and then increases linearly. With decreasing l

,
∆t decreases until the two pulses overlap when ∆t = 0. When l

decreases further,
the delay of the TM mode decreases to less than that of the constant TE mode
delay leading to increasing ∆t. Curves (c) and (d) show a linearly increasing trend
with slopes of n
o
and n
e
respectively.
Curve (e) shows the relative delay between curves (a) and (c). The relative
delay is given by
∆t
r
=
∆t
zb
−∆t
ye
∆t
zb
. (5.16)
where ∆t
zb
is ∆t for the ZDC with the input at port B (corresponding to curve
(a)) and ∆t
ye
is ∆t for the Y-junction with an input of TE mode (corresponding
to curve (c)). The decreasing trend of the relative delay suggests that, for small
108
5.3 Ti:LiNbO
3
based 1D OCDMA Code Generation
0 500 1000 1500 2000 2500 3000 3500 4000
0
50

t

(
p
s
)
0 500 1000 1500 2000 2500 3000 3500 4000
0
1
∆l (µm)
R
e
l
a
t
i
v
e

D
e
l
a
y
(e)Relative delay
between
(a)ZDC, (c)Y jn
(a)ZDC: Input at port B
(b)ZDC: Input at port A
(c)Y jn: TE mode
(d)Y jn: TM mode
Figure 5.13: Delay between the output pulses for (a) input given to port B of
ZDC, (b) input given to port A of ZDC, (c) input is only TE mode, (d) input is
only TM mode and (e) relative delay given by eqn. (5.16)
differential lengths, the ZDC with the input at port B should be preferred. For
large differential lengths, the Y-junction with an input of TE mode would give
equivalent delay.
The width of the curved and straight waveguides after the splitters (w’ and
w” in Figs. 5.9 and 5.12) can be increased or decreased to tune the refractive
indices. The widths should be no smaller than to allow low loss single mode
propagation and not large enough to bring two mode interference into the picture.
By increasing w’ to 7µm and reducing w” to 5.5µm, the delay between the two
output pulses can be increased from 40.6ps to 40.8ps.
109
Miniaturization of OCDMA Code Generation
5.4 Ti:LiNbO
3
based 2D OCDMA Code Gener-
ation
Temporal and wavelength spread 2D OCDMA codes can be generated by using
the TE-TM splitter proposed in Section 5.2 or the 3dB power splitting Y-junction.
Figure 5.14 shows a 2D OCDMA code generator which takes two input pulses and
outputs four pulses. Of the two input pulses, one has wavelength w
i
and the other
has wavelength w
j
. Each input pulse outputs two temporally spread pulses giving
a weight ‘4’ 2D OCDMA code. The wavelength pair (w
i
, w
j
) should be chosen as
per the allocation discussed in Chapters 3 and/or 4.
L
c1
L
c2
θ
TE
TM
θ
TE
S’
2w
2w’
o/p
w
w’
i/p 2
i/p 1
Figure 5.14: Miniature 2D OCDMA code generator using ZDCs
5.4.1 Design Considerations
The 2D OCDMA code generator shown in Fig. 5.14 can be accommodated on a
standard LiNbO
3
crystal by shortening the lengths of the input and output ports
of the TE-TM splitter. The necessary optimizations for the 2D code generator
over and above that of the 1D code generator follow. The gap between the upper
and lower TE-TM splitters (S’ in Fig. 5.14) should be such that input from two
different fibers can be coupled onto the Ti:LiNbO
3
device. If S’ is large, the length
of the device would increase due to the combiner at the output. The coupling
110
5.5 Ti:LiNbO
3
based 3D OCDMA Code Generation
lengths L
c1
and L
c2
would differ owing to different input wavelengths. The total
length of the TE-TM splitters should be equal in order to obtain equal delay for
both the wavelengths. Further, the waveguide widths w’ and w” (shown in Fig.
5.14) can be increased or decreased to compensate for different delays in the two
wavelengths.
5.4.2 Numerical Results
We can assume the time delay between the pulses to be the same as that of the 1D
code generator, i.e., 40ps and a chip time of 1ps, giving T = 40. For T = 40, 19 1D
OOCs of weight 2 can be generated leading to a 20 wavelength weight 4 RWOP-
based 2D MPR OCDMA code family or a 38 wavelength weight 4 CRWOP-based
2D MPR OCDMA code family. The cardinality of a 20 wavelength RWOP-based
2D code family is 168 and that of a 38 wavelength CRWOP-based 2D code family
is 361.
For T = 30, as would be in the case of the 3dB power splitting Y-junction
based 2D code generator, 14 1D OOCs of weight 2 can be generated leading
to a 14 wavelength weight 4 RWOP-based 2D MPR OCDMA code family or
a 28 wavelength weight 4 CRWOP-based 2D MPR OCDMA code family. The
cardinality of a 14 wavelength RWOP-based 2D code family is 79 and that of a
28 wavelength CRWOP-based 2D code family is 196.
5.5 Ti:LiNbO
3
based 3D OCDMA Code Gener-
ation
Temporal, wavelength and space spread 3D OCDMA codes can also be generated
by using the TE-TM splitter. Figure 5.15 shows a 3D OCDMA code generator
which takes two input pulses and outputs eight pulses. Of the two input pulses,
111
Miniaturization of OCDMA Code Generation
one has wavelength w
i
and the other has wavelength w
j
. The two input pulses
are split into the respective TE and TM modes, then combined to give 4 pulses
and the final 3dB power splitting Y-junction gives a weight ‘8’ 3D OCDMA code,
by coupling the output ports to space channels s
i
and s
j
. The wavelength pair
(w
i
, w
j
) and space channel pair (s
i
, s
j
) should be chosen as per the allocation
discussed in Chapters 3 and/or 4.
L
c1
L
c2
θ
TE
TM
θ
TE
S’
2w o/p 1
o/p 2 2w’
i/p 2
i/p 1
w’
w
Figure 5.15: Miniature 3D OCDMA code generator using ZDCs
5.5.1 Design Considerations
The 2D OCDMA code generator shown in Fig. 5.15 can be accommodated on a
standard LiNbO
3
crystal by shortening the lengths of the input and output ports
of the TE-TM splitter and/or reducing the length of the curved section. The
reduction in the length of the curved section should be equal to the length of the
3dB power splitting Y-junction. The length of the 3dB power splitting Y-junction
would be of the order of 5mm to get a separation of 125µm, which would enable
coupling of two fibers.
5.5.2 Numerical Results
Assuming the same time delay between the pulses as that of the 1D code generator,
i.e., 40ps and a chip time of 1ps, giving T = 40, the total length of the device
would be 5.5cm. For T = 40, 19 1D OOCs of weight 2 can be generated leading to
112
5.6 Insertion Losses
20 wavelength, 4 ≤ S ≤ 20 fiber, weight 8 RWOP-based 3D MPP OCDMA code
families or 38 wavelength, 4 ≤ S ≤ 38 fiber, weight 8 CRWOP-based 3D MPP
OCDMA code families. The cardinality of 20 wavelength RWOP-based 3D code
families would vary from 336 (S = 4) to 1506 (S = 20) and that of 38 wavelength
CRWOP-based 3D code families would vary from 722 (S = 4) to 6498 (S = 38).
For T = 30, as would be in the case of the 3dB power splitting Y-junction
based 3D code generator, 14 1D OOCs of weight 2 can be generated leading to
14 wavelength, 4 ≤ S ≤ 14 fiber, weight 8 RWOP-based 3D MPP OCDMA code
families or 28 wavelength, 4 ≤ S ≤ 14 fiber, weight 8 CRWOP-based 3D MPP
OCDMA code families. The cardinality of 14 wavelength RWOP-based 3D code
families would vary from 158 (S = 4) to 485 (S = 14) and that of 28 wavelength
CRWOP-based 3D code families would vary from 392 (S = 4) to 2744 (S = 28).
5.6 Insertion Losses
The power loss incurred due to the insertion of an optical device in a fiber-optic
network is known as the insertion loss of the device. The total insertion loss for the
Ti:LiNbO
3
integrated-optic devices include coupling losses and propagation losses.
The coupling losses amount to the losses encountered at the fiber-device interfaces
corresponding to the input and output ports of the device. The propagation losses
include loss due to attenuation for the length of propagation, angular bends in the
device and curved bends in the device.
For the proposed devices in sections 5.3.1, 5.3.2 and 5.4, the coupling losses
would be the same. Based on the results in [132], the coupling loss at each interface
is of the order of 1.5dB. Hence, the total coupling loss would be 3dB for each device.
From [132], the attenuation is of the order of 0.3dB per cm of propagation. For
a bend angle of 1

, the reported losses per bend are 2.6dB [133], 5dB [134]
113
Miniaturization of OCDMA Code Generation
and 0.7dB [135]. The loss of curved bends (shown in Figs. 5.9, 5.12 and 5.14)
is simulated for a constant radius of R = 15mm. The excess loss due to curved
bends is given by [136]
B = 4.34(2Γ)L

(dB), (5.17)
where 2Γ is the full width at half maximum of a resonance peak in propagation
constant versus excitation efficiency plot for the single mode curved waveguide
and L

is the length of the curved waveguide. For R = 15mm, 2Γ is 0.91 × 10
−9
for TE mode and 2.588×10
−9
for TM mode. Hence, the excess loss due to curved
bending is found to be 0.00017dB for TE mode and 0.00049dB for TM mode with
L

= 43360.3725µm.
The optical pulse attenuation for the 1D code generator based on the TE-TM
splitter (Fig. 5.9) has two parts. One corresponds to the path length including
the curved waveguide and the other corresponds to the alternate path (straight
waveguide after TE-TM splitter) from the input to the output of the device. The
length of the curved path is 5.4cm and the length of the straight path is 5.0cm.
Hence, the attenuation for the two paths would be 1.62dB and 1.5dB respectively.
Four angular bends are encountered in both the paths, which puts the bending
loss at 2.8dB [135]. The overall insertion loss for the device comes to a maximum
of 7.42dB by ignoring the low loss due to curved waveguide.
The optical pulse attenuation for the 1D code generator based on the 3dB
power splitting Y-junction (Fig. 5.12) also has the same two parts. Since the
Y-junction is shorter by about 4mm, the lengths of the two paths would be 5.0cm
and 4.6cm respectively. Thus, the attenuation for the two paths would be 1.5dB
and 1.38dB respectively. In this case, only two angular bends are encountered in
both the paths, which puts the bending loss at 1.4dB [135]. The overall insertion
loss for the device comes to a maximum of 5.9dB by ignoring the low loss due to
114
5.7 Summary
curved waveguide.
The overall insertion loss for the 2D code generator based on the TE-TM
splitter (Fig. 5.14) would increase by 0.7dB, totalling the loss to 8.12dB. The
additional 0.7dB is due to the bend near the output port where the sections for
the two different wavelengths are joined. Similarly, for a 2D code generator based
on the 3dB power splitting Y-junction, the total insertion loss would be 6.6dB.
The overall insertion loss for the 3D code generator based on the TE-TM
splitter (Fig. 5.15) would increase by a further 0.85dB, totalling the loss to 8.97dB.
Of the 0.85dB, 0.7dB is attributed to the angular bend at the junction of the 3dB
power splitting Y-junction at the output and the remaining 0.15dB is due to the
attenuation for propagation of an extra distance of 0.5cm. Similarly, for a 3D
code generator based on the 3dB power splitting Y-junction, the total insertion
loss would be 7.45dB.
From the analysis presented above, it is observed that the insertion loss of the
ZDC based devices is more than that of the 3dB power splitting Y-junction and
is higher by around 1.5dB. However, the delay performance as given in Sec. 5.3
supports the suitability of TE-TM splitter.
5.7 Summary
Refractive index profiles and propagation constants leading to the critical coupling
lengths are determined for deriving inferences to work out the dimension of the
TE-TM splitter. The quantitative results show that the number of users is lower
for the 3dB power splitting Y-junction based devices than the TE-TM splitter
based devices. The optical power calculations show lower insertion loss for the 3dB
power splitting Y-junction based devices than the TE-TM splitter based devices.
Hence, the 3dB power splitting Y-junction based devices would be suitable for
115
Miniaturization of OCDMA Code Generation
OCDMA networks where the priority is given to low loss and the TE-TM splitter
based devices would be suitable for OCDMA networks requiring larger cardinality.
A practical advantage of using the 3dB power splitting Y-junction based devices
is the abscence of polarization mode dispersion/distortion.
116
C H A P T E R 6
Conclusions and Future
Directions
6.1 Concluding Remarks
All-optical CDMA technology has the potential to offer data rates of the order of
Terabits per second. Although optical coding requires mature photonic encoder
and decoder technology, it avoids the more elaborate system design issues required
for OCDMA to emerge. For instance, encoded pulses that serve as device trig-
gers, network information carriers, or monitoring signals, may use lower bit rates
than data. This alleviates some of the physical boundaries challenging OCDMA
deployment such as dispersion and beat noise [12].
Among the various articles on code families for OCDMA, considerable empha-
sis has not been given to keep the maximum value of probability of error due to
MAI within desired limit. Existing fiber-optic communication technologies have
bit error rates of the order of 10
−8
. If the MAI itself contributes to error rates
Conclusions and Future Directions
of the order of 10
−3
, OCDMA deployment would be difficult to realize. Some
enhancements in the performance of code families and advanced integrated-optic
encoders are put forth in this thesis.
6.1.1 2D & 3D RWOP-based OCDMA Code Families
In Chapter 3, an algorithm named as row-wise orthogonal pairs (RWOP) is pro-
posed. Constructions using the RWOP algorithm and performance analysis of new
2D MPR and 3D MPP code families are given, which have a maximum crosscor-
relation value of 1 between any two codes. Out-of-phase autocorrelation value is
a maximum of 2 for 2D code families and 4 for 3D code families. The spectral
efficiency of the lower weight (2D: K

= 4 and 3D: K
′′
= 8) RWOP-based code
families is comparable to previously reported 2D and 3D code families. For higher
weight codes (2D: K

= 6 and 3D: K
′′
= 12), the spectral efficiency is observed
to be lower for RWOP-based code families. For equivalent parameters in different
2D code families, the probability of error due to MAI of the RWOP-based 2D
code families is observed to be lower by a factor of atleast 5.4 ×10
−2
as indicated
in Table 3.8. For equivalent code dimension, higher cardinality and lower weight
code families show higher probability of error due to MAI. Probability of error for
the RWOP-based 3D code families is lower than that of the SPP code families by
a factor of atleast 10
−5
as can be seen in Fig. 3.13.
6.1.2 2D & 3D CRWOP-based OCDMA Code Families
Some limitations of the RWOP algorithm like reduced cardinality as well as spec-
tral efficiency have led to the development of a new complete RWOP (CRWOP)
algorithm. The CRWOP-based 2D MPR and 3D MPP code families given in
Chapter 4 are an improvement over the RWOP-based code families. Due to the
completeness of the wavelength / space allocation algorithm, the cardinality and
118
6.1 Concluding Remarks
spectral efficiency of the CRWOP-based code families are marginally higher than
those of the RWOP-based code families at equivalent code dimension. For equiv-
alent cardinality, the code dimension of the CRWOP-based 2D code families is
lower than that of the RWOP-based 2D code families leading to better spectral
efficiency. However, the probability of error due to MAI is also marginally higher.
Comparisons based on equivalent probability of error due to MAI when all users
are interfering show that the CRWOP-based and RWOP-based code families have
lower code dimension than that of MWOOCs and SPP code families. However,
the probability of error due to MAI is higher than SPP codes when all users are
not interfering. A comparison based on equivalent code dimension shows that
the CRWOP-based 2D and 3D code families have lower probability of error at
higher spectral efficiency as compared to the RWOP-based code families. The
CRWOP-based 2D and 3D code families have lower probability of error at lower
spectral efficiency when compared with MWOOCs and SPP codes for equivalent
code dimension. The effect of CRWOP design over that of RWOP is observed in
the form of increase in the cardinality from point of view of probability of error
constrained by the desired application. The cardinalities of the proposed 2D and
3D CRWOP code families are higher by 10 and 45 respectively as compared to
those of RWOP-based 2D and 3D code families. However, the spectral efficiency
is observed to be marginally higher.
6.1.3 Miniaturization of OCDMA Code Generation
Based on some of the OCDMA code families, miniaturization of 1D, 2D and 3D
integrated-optic code generation is considered. The numerical results show that
the number of users is lower for the 3dB power splitting Y-junction based devices
than the TE-TM splitter based devices. This may be attributed to the fact that
larger the delay exhibited, higher would be the code length and thereby enhancing
119
Conclusions and Future Directions
the cardinality of user set. The optical power calculations show lower insertion
loss of around 1.5dB for the 3dB power splitting Y-junction based devices than
the TE-TM splitter based devices. Hence, the 3dB power splitting Y-junction
based devices would be suitable for OCDMA networks where the priority is given
to low loss and the TE-TM splitter based devices would be suitable for OCDMA
networks requiring larger cardinality. A practical advantage of using the 3dB
power splitting Y-junction based devices is the abscence of polarization mode
dispersion/distortion.
6.2 Scope for Further Study
The successful deployment of OCDMA needs improved code families with higher
cardinality as well as spectral efficiency without sacrificing on low probability of
error due to MAI. The CRWOP-based code families are an improvement over the
RWOP-based code families in the above sense, however some further study may
bring forth new ideas to increase the cardinality. Some improvements to the thesis
for further study may be: Improved code constructions may show higher cardi-
nality for probability of error constrained due to MAI and this may be explored.
The use of triple systems (BIBD) instead of pair-based designs in 2D and 3D code
families may be explored for enhanced spectral efficiency. In networks with inten-
sive multipath interference, the performance analysis of these code families may
be useful.
Fabrication and characterization of the proposed Titanium indiffused Lithium
Niobate encoders may bring to light some operational limitations, which in turn
could give various quantitative measurements. The measurements may provide
very useful reference to help further research. Decoders using Titanium indiffused
Lithium Niobate technology should be worked out. Such devices should be tested
120
6.2 Scope for Further Study
in a fiber-optic CDMA testbed with different code families. The link budget
analysis of a network with such integrated-optic devices would give an idea of the
feasibility of OCDMA network.
Improvements in the areas of integrated-optic short-pulse laser technology,
integrated-optic fast photo detectors and other integrated-optic devices are ex-
pected to bring revolution in the miniaturization. This will enhance the possibility
of commercial uses of OCDMA in future.
121
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Publications from the Thesis
Work
Journals (Accepted/Communicated)
1. M. Ravi Kumar, S. S. Pathak and N. B. Chakrabarti, “Design and Perfor-
mance Analysis of Code Families for Multi-Dimensional Optical CDMA”,
IET Communications vol. 3, no. 8, pp. 1311 – 1320, 2009.
2. M. Ravi Kumar, S. S. Pathak and N. B. Chakrabarti, “Design and Anal-
ysis of Three-Dimensional OCDMA Code Families”, Optical Switching and
Networking, Elsevier vol. 6, no. 4, pp. 243 – 249, 2009.
3. M. Ravi Kumar, P. Ganguly, S. S. Pathak and N. B. Chakrabarti, “Con-
struction and Generation of OCDMA Code Families using a Complete Row-
Wise Orthogonal Pairs Algorithm”, Journal of Lightwave Technology (To be
Communicated).
Conference Proceedings
1. M. Ravi Kumar, S. S. Pathak and N. B. Chakrabarti, “Design and Analysis
of New Code Families for Three-Dimensional OCDMA”, 2nd International
Symposium on Advanced Networks and Telecommunication Systems (ANTS
2008), IIT Bombay, Mumbai-India, December 15-17, 2008.
137
Publication
2. M. Ravi Kumar, P. K. Sahu, K. Esakki Muthu, P. Ganguly, S. Mahapatra
and S. S. Pathak, “Design and Analysis of Zero-Gap Directional Coupler-
Based Mode Separator”, 9th International Conference on Fiber Optics and
Photonics (Photonics 2008), IIT Delhi, Delhi-India, December 13-17 2008.
3. M. Ravi Kumar, S. S. Pathak and N. B. Chakrabarti, “A new Multi Wave-
length - Optical Code Division Multiple Access code design based on Bal-
anced Incomplete Block Design”, 2nd International Conference on Industrial
and Information Systems (ICIIS 2007), University of Peradeniya, Sri Lanka,
August 9-11, 2007.
138
Author’s Resume
M. Ravi Kumar was born in Jatni, Orissa, India in January 1979. He received his
B.E. degree from Jagannath Institute of Engineering and Technology under Utkal
University, Orissa, India in 2000 and M.Tech. degree from Cochin University of
Science and Technology, Kerala, India in 2005. He is currently working towards the
Ph. D degree at Indian Institute of Technology, Kharagpur, India. His research
interests include Optical Communication Systems, Optical Coding and Integrated-
optics.
139

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