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

(λ

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

+α

i,2

, α

i,1

+α

i,2

+α

i,3

, . . . , α

i,1

+α

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 =

2π

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

References

[1] A. Stok and E. Sargent, “System performance comparison of optical CDMA

and WDMA in a broadcast local area network,” Communications Letters,

IEEE, vol. 6, no. 9, pp. 409–411, Sep 2002.

[2] W. Huang, M. Nizam, I. Andonovic, and M. Tur, “Coherent optical CDMA

(OCDMA) systems used for high-capacity optical ﬁber networks-system de-

scription, OTDMA comparison, and OCDMA/WDMA networking,” Light-

wave Technology, Journal of, vol. 18, no. 6, pp. 765–778, Jun 2000.

[3] R. Pickholtz, L. Milstein, and D. Schilling, “Spread spectrum for mobile

communications,” Vehicular Technology, IEEE Transactions on, vol. 40,

no. 2, pp. 313 –322, may 1991.

[4] J. Y. Hui, “Pattern code modulation and optical decoding – a novel code-

division multiplexing technique for multiﬁber networks,” IEEE Journal on

Selected Areas in Communications, vol. 3, no. 6, pp. 916–927, November

1985.

[5] W. C. Kwong and P. R. Prucnal, “‘synchronous’ CDMA demonstration for

ﬁber-optic networks with optical processing,” Electronics Letters, vol. 26,

no. 24, pp. 1990–1992, November 1990.

[6] W. Kwong, P. Perrier, and P. Prucnal, “Performance comparison of asyn-

chronous and synchronous code-division multiple-access techniques for ﬁber-

optic local area networks,” Communications, IEEE Transactions on, vol. 39,

no. 11, pp. 1625 –1634, nov 1991.

[7] M. Karbassian and H. Ghafouri-Shiraz, “Performance analysis of

heterodyne-detected coherent optical CDMA using a novel prime code fam-

ily,” Lightwave Technology, Journal of, vol. 25, no. 10, pp. 3028–3034, Oct.

2007.

123

REFERENCES

[8] W. Ma, C. Zuo, H. Pu, and J. Lin, “Performance analysis on phase-encoded

OCDMA communication system,” Lightwave Technology, Journal of, vol. 20,

no. 5, pp. 798–803, May 2002.

[9] T. Hamanaka, X. Wang, N. Wada, A. Nishiki, and K. Kitayama, “Ten-user

truly asynchronous gigabit OCDMA transmission experiment with a 511-

chip SSFBG en/decoder,” Lightwave Technology, Journal of, vol. 24, no. 1,

pp. 95–102, Jan. 2006.

[10] S.-G. Park and A. Weiner, “Performance of asynchronous time-spreading

and spectrally coded OCDMA systems,” Lightwave Technology, Journal of,

vol. 26, no. 16, pp. 2873–2881, Aug.15, 2008.

[11] P. Prucnal, M. Santoro, and T. Fan, “Spread spectrum ﬁber-optic local area

network using optical processing,” Lightwave Technology, Journal of, vol. 4,

no. 5, pp. 547 – 554, may 1986.

[12] K. Fouli and M. Maier, “OCDMA and optical coding: Principles, applica-

tions, and challenges [topics in optical communications],” Communications

Magazine, IEEE, vol. 45, no. 8, pp. 27–34, August 2007.

[13] B. Ghaﬀari, M. Matinfar, and J. Salehi, “Wireless optical CDMA LAN:

digital design concepts,” Communications, IEEE Transactions on, vol. 56,

no. 12, pp. 2145–2155, December 2008.

[14] I. Andonovic, H. Sotobayashi, N. Wada, and K.-I. Kitayama, “Experimental

demonstration of the (de)coding of hybrid phase and frequency codes using

a pseudolocal oscillator for optical code division multiplexing,” Photonics

Technology Letters, IEEE, vol. 10, no. 6, pp. 887–889, Jun 1998.

[15] P. Teh, M. Ibsen, J. Lee, P. Petropoulos, and D. Richardson, “Demonstra-

tion of a four-channel WDM/OCDMA system using 255-chip 320-gchip/s

quarternary phase coding gratings,” Photonics Technology Letters, IEEE,

vol. 14, no. 2, pp. 227–229, Feb 2002.

[16] J. M. Senior, Optical Fiber Communications: Principles and Practice,

3rd ed. McGraw-Hill Education India Pvt. Ltd., Noida, India: Prentice

Hall, 2008.

[17] A. A. Shaar and P. A. Davies, “Prime sequences: quasi-optimal sequences for

OR channel code division multiplexing,” Electronics Letters, vol. 19, no. 21,

pp. 888–890, October 1983.

124

REFERENCES

[18] W. C. Kwong and P. R. Prucnal, “Ultrafast all-optical code-division

multiple-access (CDMA) ﬁber-optic networks,” Computer Networks and

ISDN Systems, vol. 26, no. 6-8, pp. 1063–1086, March 1994.

[19] J.-G. Zhang and G. Picchi, “Tunable prime-code encoder/decoder for all-

optical CDMA applications,” Electronics Letters, vol. 29, no. 13, pp. 1211–

1212, June 1993.

[20] W. Kwong, J.-G. Zhang, and G.-C. Yang, “2

n

prime-sequence code and its

optical CDMA coding architecture,” Electronics Letters, vol. 30, no. 6, pp.

509 –510, mar 1994.

[21] H. Chung and P. V. Kumar, “Optical orthogonal codes–new bounds and an

optimal construction,” IEEE Transactions on Information Theory, vol. 36,

no. 4, pp. 866–873, July 1990.

[22] S. Maric, “New family of algebraically designed optical orthogonal codes for

use in CDMA ﬁbre-optic networks,” Electronics Letters, vol. 29, no. 6, pp.

538–539, March 1993.

[23] S. Maric, Z. Kostic, and E. Titlebaum, “A new family of optical code se-

quences for use in spread-spectrum ﬁber-optic local area networks,” Com-

munications, IEEE Transactions on, vol. 41, no. 8, pp. 1217 –1221, aug

1993.

[24] G.-C. Yang and W. Kwong, “On the construction of 2

n

codes for opti-

cal code-division multiple-access,” Communications, IEEE Transactions on,

vol. 43, no. 234, pp. 495 –502, feb/mar/apr 1995.

[25] W. C. Kwong, G.-C. Yang, and J.-G. Zhang, “2

n

prime-sequence codes and

coding architecture for optical code-division multiple-access,” IEEE Trans-

actions on Communications, vol. 44, no. 9, pp. 1152–1162, September 1996.

[26] F. R. K. Chung, J. A. Salehi, and V. K. Wei, “Optical orthogonal codes:

design, analysis, and applications,” IEEE Transactions on Information The-

ory, vol. 35, no. 3, pp. 595–604, May 1989.

[27] C. Argon and R. Erg¨ ul, “Optical CDMA via shortened optical orthogonal

codes based on extended sets,” Optics Communications, vol. 116, no. 4-6,

pp. 326–330, February 1995.

[28] Y. Chang and Y. Miao, “Constructions for optimal optical orthogonal

codes,” Discrete Mathematics, vol. 261, no. 1-3, pp. 127–139, January 2003.

125

REFERENCES

[29] O. Moreno, R. Omrani, P. Kumar, and H. feng Lu, “A generalized bose-

chowla family of optical orthogonal codes and distinct diﬀerence sets,” In-

formation Theory, IEEE Transactions on, vol. 53, no. 5, pp. 1907–1910,

May 2007.

[30] J. A. Salehi, “Code division multiple-access techniques in optical ﬁber

networks–part I: fundamental principles,” IEEE Transactions on Commu-

nications, vol. 37, no. 8, pp. 824–833, August 1989.

[31] J. A. Salehi and C. A. Brackett, “Code division multiple-access techniques in

optical ﬁber networks–part II: systems performance analysis,” IEEE Trans-

actions on Communications, vol. 37, no. 8, pp. 834–842, August 1989.

[32] C. Argon, “Systematic optical orthogonal code design techniques using ex-

tended sets,” in IEEE International Conference on Communications, IEEE.

Istanbul, Turkey: IEEE, June 2006.

[33] G.-C. Yang and T. Fuja, “Optical orthogonal codes with unequal auto- and

cross-correlation constraints,” Information Theory, IEEE Transactions on,

vol. 41, no. 1, pp. 96 –106, jan. 1995.

[34] R. Fuji-Hara and Y. Miao, “Optical orthogonal codes: their bounds and new

optimal constructions,” Information Theory, IEEE Transactions on, vol. 46,

no. 7, pp. 2396 –2406, nov 2000.

[35] J. Yin, “Some combinatorial constructions for optical orthogonal codes,”

Discrete Mathematics, vol. 185, no. 1-3, pp. 201–219, April 1998.

[36] Y. Chang and J. Yin, “Further results on optimal optical orthogonal codes

with weight 4,” Discrete Mathematics, vol. 279, no. 1-3, pp. 135–151, March

2004.

[37] K. Chen and R. Wei, “A few more cyclic steiner 2-designs,” The Electronic

Journal of Combinatorics, vol. 13, no. 1, pp. 876–885, May 2006.

[38] A. Brouwer, J. Shearer, N. Sloane, and W. Smith, “A new table of constant

weight codes,” Information Theory, IEEE Transactions on, vol. 36, no. 6,

pp. 1334 –1380, nov 1990.

[39] O. Moreno, Z. Zhang, P. Kumar, and V. Zinoviev, “New constructions of

optimal cyclically permutable constant weight codes,” Information Theory,

IEEE Transactions on, vol. 41, no. 2, pp. 448 –455, mar 1995.

126

REFERENCES

[40] W. Chu and S. Golomb, “A new recursive construction for optical orthogonal

codes,” Information Theory, IEEE Transactions on, vol. 49, no. 11, pp. 3072

– 3076, nov. 2003.

[41] C. Ding and C. Xing, “Several classes of (2

m

− 1, w, 2) optical orthogonal

codes,” Discrete Applied Mathematics, vol. 128, no. 1, pp. 103–120, May

2003.

[42] W. Chu and C. J. Colbourn, “Optimal (n, 4, 2)-OOC of small orders,” Dis-

crete Mathematics, vol. 279, no. 1-3, pp. 163–172, March 2004.

[43] P. L. Check and C. J. Colbourn, “Concerning diﬀerence families with block

size four,” Discrete Mathematics, vol. 133, no. 1-3, pp. 285–289, October

1994.

[44] M. Buratti, “Constructions of (q, k, 1) diﬀerence families with q a prime

power and k = 4, 5,” Discrete Mathematics, vol. 138, no. 1-3, pp. 169–175,

March 1995.

[45] R. J. R. Abel and M. Buratti, “Some progress on (v,4,1) diﬀerence families

and optical orthogonal codes,” Journal of Combinatorial Theory, Series A,

vol. 106, no. 1, pp. 59 – 75, 2004.

[46] M. Azizoglu, J. Salehi, and Y. Li, “Optical cdma via temporal codes,” Com-

munications, IEEE Transactions on, vol. 40, no. 7, pp. 1162 –1170, jul 1992.

[47] J.-J. Chen and G.-C. Yang, “CDMA ﬁber-optic systems with optical hard

limiters,” Journal of Lightwave Technology, vol. 19, no. 7, pp. 950–958, July

2001.

[48] T. Ohtsuki, “Performance analysis of direct-detection optical asynchronous

cdma systems with double optical hard-limiters,” Lightwave Technology,

Journal of, vol. 15, no. 3, pp. 452 –457, mar 1997.

[49] S. Zahedi and J. Salehi, “Analytical comparison of various ﬁber-optic cdma

receiver structures,” Lightwave Technology, Journal of, vol. 18, no. 12, pp.

1718 –1727, dec 2000.

[50] C. Goursaud, A. Julien-Vergonjanne, C. Aupetit-Berthelemot, J.-P. Can-

ces, and J.-M. Dumas, “DS-OCDMA receivers based on parallel interfer-

ence cancellation and hard limiters,” Communications, IEEE Transactions

on, vol. 54, no. 9, pp. 1663–1671, Sept. 2006.

127

REFERENCES

[51] C.-S. Weng and J. Wu, “Optical orthogonal codes with large crosscorrelation

and their performance bound for asynchronous optical CDMA systems,”

Lightwave Technology, Journal of, vol. 21, no. 3, pp. 735–742, March 2003.

[52] M. Kumar, “Asynchronous BPPM OCDMA systems with trellis-coded mod-

ulation,” Optoelectronics, IEE Proceedings -, vol. 151, no. 4, pp. 193–201,

Aug. 2004.

[53] X. Wang, N. Wada, T. Miyazaki, G. Cincotti, and K. Kitayama, “Field trial

of 3-WDM 10-OCDMA 10.71-gb/s asynchronous WDM/DPSK-OCDMA us-

ing hybrid E/D without FEC and optical thresholding,” Lightwave Technol-

ogy, Journal of, vol. 25, no. 1, pp. 207–215, Jan. 2007.

[54] S. Mashhadi and J. Salehi, “Code-division multiple-access techniques in op-

tical ﬁber networks - part III: optical AND logic gate receiver structure with

generalized optical orthogonal codes,” Communications, IEEE Transactions

on, vol. 54, no. 8, pp. 1457–1468, Aug. 2006.

[55] X. Wang, N. Wada, G. Cincotti, T. Miyazaki, and K. Kitayama, “Demon-

stration of over 128-gb/s-capacity (12-user/spl times/10.71-gb/s/user) asyn-

chronous OCDMA using FEC and AWG-based multiport optical en-

coder/decoders,” Photonics Technology Letters, IEEE, vol. 18, no. 15, pp.

1603–1605, Aug. 2006.

[56] C.-S. Bres and P. Prucnal, “Code-empowered lightwave networks,” Light-

wave Technology, Journal of, vol. 25, no. 10, pp. 2911–2921, Oct. 2007.

[57] G.-C. Yang and W. C. Kwong, “Performance comparison of multi-

wavelength CDMA and WDMA + CDMA for ﬁber-optic networks,” IEEE

Transactions on Communications, vol. 45, no. 11, pp. 1426–1434, November

1997.

[58] K. Yu and N. Park, “Design of new family of two-dimensional wavelength-

time spreading codes for optical code division multiple access networks,”

Electronics Letters, vol. 35, no. 10, pp. 830–831, May 1999.

[59] E. Ng and E. Sargent, “Optimum threshold detection in real-time scalable

high-speed multi-wavelength optical code-division multiple-access LANs,”

Communications, IEEE Transactions on, vol. 50, no. 5, pp. 778–784, May

2002.

[60] S.-S. Lee and S.-W. Seo, “New construction of multiwavelength optical or-

thogonal codes,” Communications, IEEE Transactions on, vol. 50, no. 12,

pp. 2003 – 2008, dec 2002.

128

REFERENCES

[61] E. Shivaleela, A. Selvarajan, and T. Srinivas, “Two-dimensional optical

orthogonal codes for ﬁber-optic CDMA networks,” Lightwave Technology,

Journal of, vol. 23, no. 2, pp. 647 – 654, feb. 2005.

[62] S. Shurong, H. Yin, Z. Wang, and A. Xu, “A new family of 2-D optical

orthogonal codes and analysis of its performance in optical CDMA access

networks,” Lightwave Technology, Journal of, vol. 24, no. 4, pp. 1646–1653,

April 2006.

[63] E. S. Shivaleela and T. Srinivas, “Construction of wavelength/time codes for

ﬁber-optic CDMA networks,” IEEE Journal of Selected Topics in Quantum

Electronics, vol. 13, no. 5, pp. 1370–1377, September/October 2007.

[64] T.-W. Chang and E. Sargent, “Optimizing spectral eﬃciency in multiwave-

length optical CDMA system,” Communications, IEEE Transactions on,

vol. 51, no. 9, pp. 1442 – 1445, sept. 2003.

[65] W. C. Kwong, G.-C. Yang, V. Baby, C.-S. Br`es, and P. R. Prucnal,

“Multiple-wavelength optical orthogonal codes under prime-sequence per-

mutations for optical CDMA,” IEEE Transactions on Communications,

vol. 53, no. 1, pp. 117–123, January 2005.

[66] C.-C. Yang, J.-F. Huang, and Y.-H. Wang, “Multipulse-per-row codes for

high-speed optical wavelength/time CDMA networks,” Photonics Technol-

ogy Letters, IEEE, vol. 19, no. 21, pp. 1756–1758, Nov.1, 2007.

[67] J.-Y. Lin, J.-S. Jhou, C.-Y. Liu, and J.-H. Wen, “Performance analysis of

modiﬁed prime-hop codes for OCDMA systems with multiuser detectors,”

Optical Fiber Technology, vol. 13, no. 2, pp. 108 – 116, 2007.

[68] C.-C. Yang, “Optical CDMA coding scheme with a large size of code space,”

Communications Letters, IEEE, vol. 13, no. 2, pp. 145–147, February 2009.

[69] M. Karbassian and H. Ghafouri-Shiraz, “Incoherent two-dimensional array

modulation transceiver for photonic CDMA,” Lightwave Technology, Jour-

nal of, vol. 27, no. 8, pp. 980–988, April15, 2009.

[70] F.-R. Gu and J. Wu, “Construction of two-dimensional wavelength/time op-

tical orthogonal codes using diﬀerence family,” Lightwave Technology, Jour-

nal of, vol. 23, no. 11, pp. 3642–3652, Nov. 2005.

[71] H. Cao and R. Wei, “Combinatorial constructions for optimal two-

dimensional optical orthogonal codes,” Information Theory, IEEE Trans-

actions on, vol. 55, no. 3, pp. 1387–1394, March 2009.

129

REFERENCES

[72] T. Bazan, D. Harle, and I. Andonovic, “Code ﬂexibility of 2-D time-

spreading wavelength-hopping in OCDMA systems,” Selected Topics in

Quantum Electronics, IEEE Journal of, vol. 13, no. 5, pp. 1378–1385, Sept.-

oct. 2007.

[73] S. Galli, R. Menendez, E. Narimanov, and P. Prucnal, “A novel method for

increasing the spectral eﬃciency of optical CDMA,” Communications, IEEE

Transactions on, vol. 56, no. 12, pp. 2133 –2144, december 2008.

[74] T.-W. Chang and E. Sargent, “Optical CDMA using 2-D codes: the optimal

single-user detector,” Communications Letters, IEEE, vol. 5, no. 4, pp. 169

–171, apr 2001.

[75] J. Faucher, S. Ayotte, L. Rusch, S. LaRochelle, and D. Plant, “Experimental

BER performance of 2D λ-t OCDMA with recovered clock,” Electronics

Letters, vol. 41, no. 12, pp. 713–715, June 2005.

[76] J. Faucher, S. Ayotte, Z. El-Sahn, M. Mukadam, L. Rusch, and D. Plant,

“A standalone receiver with multiple access interference rejection, clock and

data recovery, and FEC for 2-D OCDMA,” Photonics Technology Letters,

IEEE, vol. 18, no. 20, pp. 2123–2125, Oct. 2006.

[77] E. S. Shivaleela, K. N. Sivarajan, and A. Selvarajan, “Design of a new fam-

ily of two-dimensional codes for ﬁber-optic CDMA networks,” Journal of

Lightwave Technology, vol. 16, no. 4, pp. 501–508, April 1998.

[78] C.-C. Hsu, G.-C. Yang, and W. Kwong, “Hard-limiting performance analysis

of 2-d optical codes under the chip-asynchronous assumption,” Communi-

cations, IEEE Transactions on, vol. 56, no. 5, pp. 762 –768, may 2008.

[79] F. Uherek and J. Chovan, “2-d wavelength-time optical CDMA system -

experiment and simulation,” in Transparent Optical Networks, 2007. ICTON

’07. 9th International Conference on, vol. 1, July 2007, pp. 118–121.

[80] I. Glesk, P. Prucnal, and I. Andonovic, “Incoherent ultrafast OCDMA re-

ceiver design with 2 ps all-optical time gate to suppress multiple-access inter-

ference,” Selected Topics in Quantum Electronics, IEEE Journal of, vol. 14,

no. 3, pp. 861–867, May-june 2008.

[81] K. Takiguchi, H. Takahashi, O. Moriwaki, and A. Okuno, “Integrated pho-

tonic decoder with complementary code processing and balanced detection

for two-dimensional OCDMA,” in Optical Fiber Communication and the Na-

tional Fiber Optic Engineers Conference, 2007. OFC/NFOEC 2007. Con-

ference on, March 2007, pp. 1–3.

130

REFERENCES

[82] A. A. Garba and J. Bajcsy, “A new approach to achieve high spectral eﬃ-

ciency in wavelength-time OCDMA network transmission,” Photonics Tech-

nology Letters, IEEE, vol. 19, no. 3, pp. 131–133, Feb.1, 2007.

[83] C.-M. Tsai, “Optical wavelength/spatial coding system based on quadratic

congruence code matrices,” Photonics Technology Letters, IEEE, vol. 18,

no. 17, pp. 1843–1845, Sept. 2006.

[84] C.-C. Yang, J.-F. Huang, and I.-M. Chiu, “Performance analyses on hybrid

MQC/M-sequence coding over frequency/spatial optical CDMA system,”

Communications, IEEE Transactions on, vol. 55, no. 1, pp. 40–43, Jan.

2007.

[85] Z. Zhang, C. Tian, M. Mokhtar, P. Petropoulos, D. Richardson, and M. Ib-

sen, “Rapidly reconﬁgurable optical phase encoder-decoders based on ﬁber

bragg gratings,” Photonics Technology Letters, IEEE, vol. 18, no. 11, pp.

1216–1218, June 2006.

[86] V. Hernandez, Y. Du, W. Cong, R. Scott, K. Li, J. Heritage, Z. Ding,

B. Kolner, and S. Yoo, “Spectral phase-encoded time-spreading (SPECTS)

optical code-division multiple access for terabit optical access networks,”

Lightwave Technology, Journal of, vol. 22, no. 11, pp. 2671–2679, Nov. 2004.

[87] Y. Du, J. Cao, and S. Yoo, “Performance comparison of gated and nongated

all-optical thresholding detection schemes using machzehnder interferome-

ters in SPECTS O-CDMA,” Photonics Technology Letters, IEEE, vol. 19,

no. 14, pp. 1054–1056, July15, 2007.

[88] T. Miyazawa and I. Sasase, “Multirate spectral phase-encoded time-

spreading O-CDMA system using orthogonal variable spreading factor code

sequences,” Photonics Technology Letters, IEEE, vol. 19, no. 19, pp. 1502–

1504, Oct.1, 2007.

[89] X. Wang, “Novel time domain spectral phase encoding/decoding technique

for OCDMA application,” in Transparent Optical Networks, 2009. ICTON

’09. 11th International Conference on, 28 2009-July 2 2009, pp. 1–4.

[90] R. Omrani and P. Kumar, “Spreading sequences for asynchronous spectrally

phase encoded optical CDMA,” in Information Theory, 2006 IEEE Inter-

national Symposium on, July 2006, pp. 2642–2646.

[91] M. Rochette and L. Rusch, “Spectral eﬃciency of OCDMA systems with

coherent pulsed sources,” Lightwave Technology, Journal of, vol. 23, no. 3,

pp. 1033–1038, March 2005.

131

REFERENCES

[92] A. Agarwal, P. Toliver, R. Menendez, T. Banwell, J. Jackel, and S. Etemad,

“Spectrally eﬃcient six-user coherent OCDMA system using reconﬁgurable

integrated ring resonator circuits,” Photonics Technology Letters, IEEE,

vol. 18, no. 18, pp. 1952–1954, Sept.15, 2006.

[93] S. Khaleghi, S. Khaleghi, and K. Jamshidi, “Performance analysis of a spec-

trally phase-encoded optical code division multiple access packet network,”

Optical Communications and Networking, IEEE/OSA Journal of, vol. 1,

no. 3, pp. 213–221, August 2009.

[94] Z. Zhang, C. Tian, P. Petropoulos, D. Richardson, and M. Ibsen,

“Distributed-phase OCDMA encoder decoders based on ﬁber bragg grat-

ings,” Photonics Technology Letters, IEEE, vol. 19, no. 8, pp. 574–576,

April15, 2007.

[95] C. Chua, F. Abbou, H. Chuah, and S. Majumder, “Performance analysis on

phase-encoded OCDMA communication system in dispersive ﬁber medium,”

Photonics Technology Letters, IEEE, vol. 16, no. 2, pp. 668–670, Feb. 2004.

[96] S. Etemad, P. Toliver, R. Menendez, J. Young, T. Banwell, S. Galli, J. Jackel,

P. Delfyett, C. Price, and T. Turpin, “Spectrally eﬃcient optical CDMA us-

ing coherent phase-frequency coding,” Photonics Technology Letters, IEEE,

vol. 17, no. 4, pp. 929–931, April 2005.

[97] R. Menendez, P. Toliver, S. Galli, A. Agarwal, T. Banwell, J. Jackel,

J. Young, and S. Etemad, “Network applications of cascaded passive code

translation for WDM-compatible spectrally phase-encoded optical CDMA,”

Lightwave Technology, Journal of, vol. 23, no. 10, pp. 3219–3231, Oct. 2005.

[98] P. Teh, M. Ibsen, and D. Richardson, “Demonstration of a full-duplex bidi-

rectional spectrally interleaved OCDMA/DWDM system,” Photonics Tech-

nology Letters, IEEE, vol. 15, no. 3, pp. 482–484, March 2003.

[99] X. Wang, N. Wada, T. Hamanaka, T. Miyazaki, G. Cincotti, and K. Ki-

tayama, “OCDMA over WDM transmission,” in Transparent Optical Net-

works, 2007. ICTON ’07. 9th International Conference on, vol. 1, july 2007,

pp. 110 –113.

[100] C. Tian, Z. Zhang, M. Ibsen, P. Petropoulos, and D. Richardson, “A 16-

channel reconﬁgurable OCDMA/DWDM system using continuous phase-

shift SSFBGs,” Selected Topics in Quantum Electronics, IEEE Journal of,

vol. 13, no. 5, pp. 1480–1486, Sept.-oct. 2007.

132

REFERENCES

[101] N. Kataoka, N. Wada, X. Wang, G. Cincotti, A. Sakamoto, Y. Terada,

T. Miyazaki, and K.-i. Kitayama, “Field trial of duplex, 10 gbps ×8-user

DPSK-OCDMA system using a single 16× 16 multi-port encoder/decoder

and 16-level phase-shifted SSFBG encoder/decoders,” Lightwave Technol-

ogy, Journal of, vol. 27, no. 3, pp. 299–305, Feb.1, 2009.

[102] S. Kim, K. Yu, and N. Park, “A new family of space/wavelength/time spread

three-dimensional optical code for OCDMA networks,” Lightwave Technol-

ogy, Journal of, vol. 18, no. 4, pp. 502–511, Apr 2000.

[103] J. McGeehan, S. Nezam, P. Saghari, A. Willner, R. Omrani, and P. Ku-

mar, “Experimental demonstration of OCDMA transmission using a three-

dimensional (time-wavelength-polarization) codeset,” Lightwave Technology,

Journal of, vol. 23, no. 10, pp. 3282–3289, Oct. 2005.

[104] S. Jindal and N. Gupta, “Performance evaluation of optical CDMA based

3D code with increasing bit rate in local area network,” in Computational

Technologies in Electrical and Electronics Engineering, 2008. SIBIRCON

2008. IEEE Region 8 International Conference on, July 2008, pp. 386–388.

[105] B.-C. Yeh, C.-H. Lin, and J. Wu, “Noncoherent spectral/time/spatial optical

CDMA system using 3-D perfect diﬀerence codes,” Lightwave Technology,

Journal of, vol. 27, no. 6, pp. 744–759, March15, 2009.

[106] J. Singh and M. L. Singh, “A new family of three-dimensional codes for op-

tical CDMA systems with diﬀerential detection,” Optical Fiber Technology,

vol. 15, no. 5-6, pp. 470 – 476, 2009.

[107] J. Singh and M. Singh, “Design of 3-D wavelength/time/space codes for

asynchronous ﬁber-optic CDMA systems,” Photonics Technology Letters,

IEEE, vol. 22, no. 3, pp. 131 –133, feb.1, 2010.

[108] M. Morelle, C. Goursaud, A. Julien-Vergonjanne, C. Aupetit-Berthelemot,

J.-P. Cances, J.-M. Dumas, and P. Guignard, “2-dimensional optical cdma

system performance with parallel interference cancellation,” Computers and

Communications, IEEE Symposium on, vol. 0, pp. 634–640, 2006.

[109] R. A. Brualdi, Introductory Combinatorics, 1st ed. North-Holland, New

York: North-Holland, 1977.

[110] R. C. Bose, “On the construction of balanced incomplete block designs,”

Annals of Eugenics, vol. 9, pp. 353–399, 1939.

133

REFERENCES

[111] R. M. Wilson, “Cyclotomy and diﬀerence families in elementary abelian

groups,” Journal of Number Theory, vol. 4, no. 1, pp. 17–47, February 1972.

[112] C. J. Colbourn and A. Rosa, Triple Systems, 1st ed. Oxford University

Press, Oxford: Clarendon Press, 1999.

[113] Y. Silberberg, P. Perlmutter, and J. E. Baran, “Digital optical switch,”

Applied Physics Letters, vol. 51, no. 16, pp. 1230 –1232, oct 1987.

[114] E. Wooten, K. Kissa, A. Yi-Yan, E. Murphy, D. Lafaw, P. Hallemeier,

D. Maack, D. Attanasio, D. Fritz, G. McBrien, and D. Bossi, “A review

of lithium niobate modulators for ﬁber-optic communications systems,” Se-

lected Topics in Quantum Electronics, IEEE Journal of, vol. 6, no. 1, pp. 69

–82, jan/feb 2000.

[115] C. Y. Huang, C. H. Lin, Y. H. Chen, and Y. C. Huang, “Electro-optic

ti:ppln waveguide as eﬃcient optical wavelength ﬁlter and polarization mode

converter,” Opt. Express, vol. 15, no. 5, pp. 2548–2554, Mar 2007. [Online].

Available: http://www.opticsexpress.org/abstract.cfm?URI=oe-15-5-2548

[116] W. Huang and C. Xu, “Simulation of three-dimensional optical waveguides

by a full-vector beam propagation method,” Quantum Electronics, IEEE

Journal of.

[117] K. Kawano and T. Kitoh, Beam Propagation Methods. John

Wiley & Sons, Inc., 2002, pp. 165–231. [Online]. Available:

http://dx.doi.org/10.1002/0471221600.ch5

[118] A. Ghatak, K. Thyagarajan, and M. Shenoy, “Numerical analysis of planar

optical waveguides using matrix approach,” Lightwave Technology, Journal

of, vol. 5, no. 5, pp. 660 – 667, may 1987.

[119] P. Ganguly, D. C. Sen, S. Datt, J. C. Biswas, and S. K. Lahiri, “Simulation

of refractive index proﬁles for titanium indiﬀused lithium niobate channel

waveguides,” Fiber and Integrated Optics, vol. 15, no. 2, pp. 135–147, 1996.

[120] P. Ganguly, J. C. Biswas, and S. K. Lahiri, “Matrix-based analytical model

of critical coupling length of titanium in-diﬀused integrated-optic directional

coupler on lithium niobate substrate,” Fiber and Integrated Optics, vol. 17,

no. 2, pp. 139–155, 1998.

[121] E. S. Shivaleela, “Design and performance analysis of a new family of wave-

length/time codes for ﬁber-optic CDMA networks,” Ph.D. dissertation, IISc,

Division of Electrical Sciences, Electrical Communication Engineering, Ban-

galore, India, Nov. 2007.

134

REFERENCES

[122] P. Ganguly, J. C. Biswas, and S. K. Lahiri, “Analysis of Ti:LiNbO

3

zero-

gap directional coupler for wavelength division multiplexer/demultiplexer,”

Optics Communications, vol. 281, no. 12, pp. 3269 – 3274, 2008.

[123] F. Rottmann, A. Neyer, W. Mevenkamp, and E. Voges, “Integrated-optic

wavelength multiplexers on lithium niobate based on two-mode interfer-

ence,” Lightwave Technology, Journal of, vol. 6, no. 6, pp. 946 –952, jun

1988.

[124] H. Nishihara, M. Haruna, and T. Suhara, Optical integrated circuits, 1st ed.

McGraw-Hill Book Co., New York: Mc-Graw Hill, 1989.

[125] A. Ghatak, Optics, 4th ed. McGraw-Hill Education India Pvt. Ltd., Noida,

India: Tata McGraw-Hill, 2008.

[126] N. Goto and G. Yip, “A te-tm mode splitter in LiNbO

3

by proton exchange

and Ti diﬀusion,” Lightwave Technology, Journal of, vol. 7, no. 10, pp. 1567–

1574, Oct 1989.

[127] P.-K. Wei and W.-S. Wang, “A TE-TM mode splitter on lithium niobate

using Ti, Ni, and MgO diﬀusions,” Photonics Technology Letters, IEEE,

vol. 6, no. 2, pp. 245–248, Feb 1994.

[128] W.-H. Hsu, K.-C. Lin, J.-Y. Li, Y.-S. Wu, and W.-S. Wang, “Polarization

splitter with variable TE-TM mode converter using Zn and Ni codiﬀused

LiNbO

3

waveguides,” Selected Topics in Quantum Electronics, IEEE Jour-

nal of, vol. 11, no. 1, pp. 271–277, Jan.-Feb. 2005.

[129] W. Zheng, M. Xing, G. Ren, S. G. Johnson, W. Zhou, W. Chen, and L. Chen,

“Integration of a photonic crystal polarization beam splitter and waveguide

bend,” Opt. Express, vol. 17, no. 10, pp. 8657–8668, 2009.

[130] H. Fukuda, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Shinojima, and

S. ichi Itabashi, “Silicon photonic circuit with polarization diversity,” Opt.

Express, vol. 16, no. 7, pp. 4872–4880, 2008.

[131] R. Chakraborty, P. Ganguly, J. C. Biswas, and S. K. Lahiri, “Modal pro-

ﬁles in Ti:LiNbO

3

two-waveguide and three-waveguide couplers by eﬀective-

index-based matrix method,” Optics Communications, vol. 187, no. 1-3, pp.

155 – 163, 2001.

[132] P. Ganguly, B. Umapathi, S. Das, J. C. Biswas, and S. K. Lahiri, “Fabri-

cation and characterization of ti:linbo

3

waveguides,” in Proceedings of In-

ternational Conference on Optics and Opto-electronics, December 1998, pp.

pp.450–456.

135

REFERENCES

[133] M. J. Taylor and E. R. Schumacher, “Measured losses in linbo3 waveguide

bends,” Appl. Opt., vol. 19, no. 18, pp. 3048–3050, 1980.

[134] L. D. Hutcheson, I. A. White, and J. J. Burke, “Comparison of bending

losses in integrated optical circuits,” Opt. Lett., vol. 5, no. 6, pp. 276–278,

1980.

[135] R.-C. Lu, Y.-P. Liao, H.-B. Lin, and W.-S. Wang, “Design and fabrication of

wide-angle abrupt bends on lithium niobate,” Selected Topics in Quantum

Electronics, IEEE Journal of, vol. 2, no. 2, pp. 215 –220, jun 1996.

[136] P. Ganguly, J. C. Biswas, and S. K. Lahiri, “Modelling of titanium indif-

fused lithium niobate channel waveguide bends: a matrix approach,” Optics

Communications, vol. 155, no. 1-3, pp. 125 – 134, 1998.

136

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